InverseBetaRegularized[z, a, b] \[Proportional] InverseBetaRegularized[z, Subscript[a, 0], b] - (1 - w)^(1 - b) w ((-(1/Subscript[a, 0]^2)) HypergeometricPFQ[ {Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] + (Beta[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[ Subscript[a, 0] + b]))/w^Subscript[a, 0]) (a - Subscript[a, 0]) + (1/2) (1 - w)^(1 - 2 b) w (-2 (1 - w)^b Gamma[Subscript[a, 0]]^3 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - ((1 - w)^b Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (PolyGamma[Subscript[a, 0]] - PolyGamma[Subscript[a, 0] + b]) (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b]))/ w^Subscript[a, 0] + (-1 + b) w^(1 - Subscript[a, 0]) Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[ {Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] (w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b])) + ((-1 + w) Gamma[Subscript[a, 0]]^2 (-HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] + Subscript[a, 0]^2 (-1 + b) w HypergeometricPFQRegularized[ {1 + Subscript[a, 0], 1 + Subscript[a, 0], 2 - b}, {2 + Subscript[a, 0], 2 + Subscript[a, 0]}, w]) (w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b])))/w^Subscript[a, 0] + (1 - b) w^(1 - 2 Subscript[a, 0]) Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b]) (w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b])) - ((1 - w)^b Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b]) (-Log[w] + ((1 - Subscript[a, 0]) (1 - w)^(1 - b) (w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b])))/w^Subscript[a, 0]))/w^Subscript[a, 0] + (Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] ((-1 + w) w^Subscript[a, 0] Gamma[Subscript[a, 0]]^2 HypergeometricPFQRegularized[{Subscript[a, 0], Subscript[a, 0], 1 - b}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, w] - (-1 + w) Beta[Subscript[a, 0], b] BetaRegularized[w, Subscript[a, 0], b] (Log[w] - PolyGamma[Subscript[a, 0]] + PolyGamma[Subscript[a, 0] + b]) + (1 - w)^b w^Subscript[a, 0] (PolyGamma[1, Subscript[a, 0]] - PolyGamma[1, Subscript[a, 0] + b])))/w^(2 Subscript[a, 0])) (a - Subscript[a, 0])^2 + \[Ellipsis] /; (a -> Subscript[a, 0]) && w == InverseBetaRegularized[z, Subscript[a, 0], b]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], "-", " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", "b"]]], " ", "w", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", SubsuperscriptBox["a", "0", "2"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "+", RowBox[List[SuperscriptBox["w", RowBox[List["-", SubscriptBox["a", "0"]]]], " ", RowBox[List["Beta", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]], RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]]], " ", "w", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], "3"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox["w", RowBox[List["-", SubscriptBox["a", "0"]]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ")"]], " ", SuperscriptBox["w", RowBox[List["1", "-", SubscriptBox["a", "0"]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["w", SubscriptBox["a", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]]]], ")"]], "+", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", SuperscriptBox["w", RowBox[List["-", SubscriptBox["a", "0"]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "+", RowBox[List[SuperscriptBox[SubscriptBox["a", "0"], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ")"]], " ", "w", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["2", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["w", SubscriptBox["a", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]]]], ")"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", SuperscriptBox["w", RowBox[List["1", "-", RowBox[List["2", " ", SubscriptBox["a", "0"]]]]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["w", SubscriptBox["a", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox["w", RowBox[List["-", SubscriptBox["a", "0"]]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "w", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["a", "0"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", "b"]]], " ", SuperscriptBox["w", RowBox[List["-", SubscriptBox["a", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["w", SubscriptBox["a", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["w", RowBox[List[RowBox[List["-", "2"]], " ", SubscriptBox["a", "0"]]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", SuperscriptBox["w", SubscriptBox["a", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["a", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["a", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["a", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["a", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["a", "0"], "+", "b"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox["w", SubscriptBox["a", "0"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", SubscriptBox["a", "0"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[SubscriptBox["a", "0"], "+", "b"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["a", "\[Rule]", SubscriptBox["a", "0"]]], ")"]], "\[And]", RowBox[List["w", "\[Equal]", RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", SubscriptBox["a", "0"], ",", "b"]], "]"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <msubsup> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mtext> </mtext> <mrow> <mi> w </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> w </mi> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msubsup> <mi> a </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mi> F </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> w </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 4 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;4&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;3&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mi> w </mi> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> </mrow> <mo> - </mo> <mrow> <msup> <mi> w </mi> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> w </mi> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> w </mi> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> w </mi> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mrow> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msubsup> <mi> a </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> w </mi> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;2&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;2&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> w </mi> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> w </mi> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> w </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> w </mi> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 3 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 2 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ; </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;3&quot;, TraditionalForm]], SubscriptBox[OverscriptBox[&quot;F&quot;, &quot;~&quot;], FormBox[&quot;2&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[SubscriptBox[&quot;a&quot;, &quot;0&quot;], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[SubscriptBox[&quot;a&quot;, &quot;0&quot;], &quot;+&quot;, &quot;1&quot;]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, False]], &quot;;&quot;, TagBox[&quot;w&quot;, HypergeometricPFQRegularized, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQRegularized] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> w </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> w </mi> <mo> &#10869; </mo> <mrow> <msubsup> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> InverseBetaRegularized </ci> <ci> z </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <plus /> <apply> <ci> InverseBetaRegularized </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> w </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <ci> Beta </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <ci> w </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> w </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <ci> w </ci> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </list> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </list> <ci> w </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <apply> <ci> Beta </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <ci> BetaRegularized </ci> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> <apply> <plus /> <apply> <ln /> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <eq /> <ci> w </ci> <apply> <ci> InverseBetaRegularized </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseBetaRegularized", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", "b"]]], " ", "w", " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]], SubsuperscriptBox["aa", "0", "2"]]]], "+", RowBox[List[SuperscriptBox["w", RowBox[List["-", SubscriptBox["aa", "0"]]]], " ", RowBox[List["Beta", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]]], " ", "w", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], "3"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox["w", RowBox[List["-", SubscriptBox["aa", "0"]]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ")"]], " ", SuperscriptBox["w", RowBox[List["1", "-", SubscriptBox["aa", "0"]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["w", SubscriptBox["aa", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", SuperscriptBox["w", RowBox[List["-", SubscriptBox["aa", "0"]]]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "+", RowBox[List[SubsuperscriptBox["aa", "0", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ")"]], " ", "w", " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["2", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["2", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["w", SubscriptBox["aa", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", SuperscriptBox["w", RowBox[List["1", "-", RowBox[List["2", " ", SubscriptBox["aa", "0"]]]]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["w", SubscriptBox["aa", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox["w", RowBox[List["-", SubscriptBox["aa", "0"]]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "w", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["aa", "0"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", "b"]]], " ", SuperscriptBox["w", RowBox[List["-", SubscriptBox["aa", "0"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["w", SubscriptBox["aa", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["w", RowBox[List[RowBox[List["-", "2"]], " ", SubscriptBox["aa", "0"]]]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", SuperscriptBox["w", SubscriptBox["aa", "0"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], "2"], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "w"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", RowBox[List["Beta", "[", RowBox[List[SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["BetaRegularized", "[", RowBox[List["w", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "w", "]"]], "-", RowBox[List["PolyGamma", "[", SubscriptBox["aa", "0"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox["w", SubscriptBox["aa", "0"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", SubscriptBox["aa", "0"]]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List[SubscriptBox["aa", "0"], "+", "b"]]]], "]"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["a", "\[Rule]", SubscriptBox["aa", "0"]]], ")"]], "&&", RowBox[List["w", "\[Equal]", RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]]]]]]]]]]]]

2007-05-02