html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HypergeometricPFQ

 http://functions.wolfram.com/07.27.06.0040.01

 Input Form

 HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == Subscript[F, Infinity][z, Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[b, 1], Subscript[b, 2]] /; (Subscript[F, n][z, Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[b, 1], Subscript[b, 2]] == ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]])) ((((Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2] - Subscript[a, 1]] Gamma[Subscript[a, 3] - Subscript[a, 1]])/ (Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]])) Sum[(Pochhammer[Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 1] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 1] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 1] - Subscript[a, 2], k] Pochhammer[1 + Subscript[a, 1] - Subscript[a, 3], k]), {k, 0, n}])/(-z)^Subscript[a, 1] + (((Gamma[Subscript[a, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[Subscript[a, 3] - Subscript[a, 2]])/ (Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]])) Sum[(Pochhammer[Subscript[a, 2], k] Pochhammer[1 + Subscript[a, 2] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 2] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 2] - Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 2] - Subscript[a, 3], k]), {k, 0, n}])/(-z)^Subscript[a, 2] + (((Gamma[Subscript[a, 3]] Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]])/ (Gamma[Subscript[b, 1] - Subscript[a, 3]] Gamma[Subscript[b, 2] - Subscript[a, 3]])) Sum[(Pochhammer[Subscript[a, 3], k] Pochhammer[1 + Subscript[a, 3] - Subscript[b, 1], k] Pochhammer[1 + Subscript[a, 3] - Subscript[b, 2], k])/z^k/(k! Pochhammer[1 + Subscript[a, 3] - Subscript[a, 1], k] Pochhammer[1 + Subscript[a, 3] - Subscript[a, 2], k]), {k, 0, n}])/(-z)^Subscript[a, 3]) == HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] - ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]])) (((Sin[(Subscript[b, 1] - Subscript[a, 1]) Pi] Sin[(Subscript[b, 2] - Subscript[a, 1]) Pi] Gamma[1 + Subscript[a, 1] + n] Gamma[2 + Subscript[a, 1] - Subscript[b, 1] + n] Gamma[2 + Subscript[a, 1] - Subscript[b, 2] + n] z^(-n - 1))/(-z)^Subscript[a, 1]/ (Sin[(Subscript[a, 2] - Subscript[a, 1]) Pi] Sin[(Subscript[a, 3] - Subscript[a, 1]) Pi] Gamma[2 + Subscript[a, 1] - Subscript[a, 2] + n] Gamma[2 + Subscript[a, 1] - Subscript[a, 3] + n] (n + 1)!)) HypergeometricPFQ[{1, 1 + Subscript[a, 1] + n, 2 + Subscript[a, 1] - Subscript[b, 1] + n, 2 + Subscript[a, 1] - Subscript[b, 2] + n}, {2 + n, 2 + Subscript[a, 1] - Subscript[a, 2] + n, 2 + Subscript[a, 1] - Subscript[a, 3] + n}, 1/z] + ((Sin[(Subscript[b, 1] - Subscript[a, 2]) Pi] Sin[(Subscript[b, 2] - Subscript[a, 2]) Pi] Gamma[1 + Subscript[a, 2] + n] Gamma[2 + Subscript[a, 2] - Subscript[b, 1] + n] Gamma[2 + Subscript[a, 2] - Subscript[b, 2] + n] z^(-n - 1))/(-z)^Subscript[a, 2]/ (Sin[(Subscript[a, 1] - Subscript[a, 2]) Pi] Sin[(Subscript[a, 3] - Subscript[a, 2]) Pi] Gamma[2 + Subscript[a, 2] - Subscript[a, 1] + n] Gamma[2 + Subscript[a, 2] - Subscript[a, 3] + n] (n + 1)!)) HypergeometricPFQ[{1, 1 + Subscript[a, 2] + n, 2 + Subscript[a, 2] - Subscript[b, 1] + n, 2 + Subscript[a, 2] - Subscript[b, 2] + n}, {2 + n, 2 + Subscript[a, 2] - Subscript[a, 1] + n, 2 + Subscript[a, 2] - Subscript[a, 3] + n}, 1/z] + ((Sin[(Subscript[b, 1] - Subscript[a, 3]) Pi] Sin[(Subscript[b, 2] - Subscript[a, 3]) Pi] Gamma[1 + Subscript[a, 3] + n] Gamma[2 + Subscript[a, 3] - Subscript[b, 1] + n] Gamma[2 + Subscript[a, 3] - Subscript[b, 2] + n] z^(-n - 1))/(-z)^Subscript[a, 3]/ (Sin[(Subscript[a, 2] - Subscript[a, 3]) Pi] Sin[(Subscript[a, 1] - Subscript[a, 3]) Pi] Gamma[2 + Subscript[a, 3] - Subscript[a, 2] + n] Gamma[2 + Subscript[a, 3] - Subscript[a, 1] + n] (n + 1)!)) HypergeometricPFQ[{1, 1 + Subscript[a, 3] + n, 2 + Subscript[a, 3] - Subscript[b, 1] + n, 2 + Subscript[a, 3] - Subscript[b, 2] + n}, {2 + n, 2 + Subscript[a, 3] - Subscript[a, 2] + n, 2 + Subscript[a, 3] - Subscript[a, 1] + n}, 1/z]) && Element[n, Integers] && n >= 0) && !Element[Subscript[a, 1] - Subscript[a, 2], Integers] && !Element[Subscript[a, 1] - Subscript[a, 3], Integers] && !Element[Subscript[a, 2] - Subscript[a, 3], Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"], ",", SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["a", "2"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["a", "3"], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "1"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["a", "1"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "1"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "1"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"]]], ",", "k"]], "]"]], SuperscriptBox["z", RowBox[List["-", "k"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]], ",", "k"]], "]"]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["a", "2"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "2"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "2"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "2"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"]]], ",", "k"]], "]"]], SuperscriptBox["z", RowBox[List["-", "k"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["a", "1"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]], ",", "k"]], "]"]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "3"], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "3"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"]]], ",", "k"]], "]"]], SuperscriptBox["z", RowBox[List["-", "k"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["k", "!"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["a", "1"]]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "3"], "-", SubscriptBox["a", "2"]]], ",", "k"]], "]"]]]], ")"]]]]]]]]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", SubscriptBox["a", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"]]], "}"]], ",", "z"]], "]"]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["b", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["b", "2"], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "1"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["a", "2"], "]"]], RowBox[List["Gamma", "[", SubscriptBox["a", "3"], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "1"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "1"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["a", "1"], "+", "n"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "n"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "1"]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "1"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["a", "1"]]], ")"]], "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"], "+", "n"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]], ")"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["a", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "1"], "-", SubscriptBox["b", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"], "+", "n"]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]], "+", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "2"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "2"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["a", "2"], "+", "n"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "n"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "2"]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "3"], "-", SubscriptBox["a", "2"]]], ")"]], "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "2"], "-", SubscriptBox["a", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", "n"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]], ")"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["a", "2"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "2"], "-", SubscriptBox["b", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "2"], "-", SubscriptBox["a", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"], "+", "n"]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]], "+", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "1"], "-", SubscriptBox["a", "3"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["b", "2"], "-", SubscriptBox["a", "3"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["a", "3"], "+", "n"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "n"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["a", "3"]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]], ")"]], "\[Pi]"]], "]"]], RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]], ")"]], "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "3"], "-", SubscriptBox["a", "2"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["a", "3"], "-", SubscriptBox["a", "1"], "+", "n"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]], ")"]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["a", "3"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "3"], "-", SubscriptBox["b", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "3"], "-", SubscriptBox["a", "2"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["a", "3"], "-", SubscriptBox["a", "1"], "+", "n"]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]]]], ")"]]]]]]]], StyleBox[")", Rule[FontWeight, "Plain"]]]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "2"]]], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "-", SubscriptBox["a", "3"]]], ",", "Integers"]], "]"]], "]"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "-", SubscriptBox["a", "3"]]], ",", "Integers"]], "]"]], "]"]]]]]]]]

 MathML Form

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", SubscriptBox["a_", "2"], ",", SubscriptBox["a_", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", SubscriptBox["b_", "2"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", SubscriptBox["aa", "3"], ",", SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"]]], "]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", SubscriptBox["aa", "3"], ",", SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"]]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "1"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "1"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["aa", "1"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "1"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["bb", "1"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["bb", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "3"]]], ",", "k"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "1"]]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "2"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["aa", "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "2"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["bb", "1"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["bb", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "1"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "3"]]], ",", "k"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "2"]]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "3"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "3"]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["aa", "3"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["aa", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["bb", "1"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["bb", "2"]]], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "1"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "2"]]], ",", "k"]], "]"]]]]]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "3"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "3"]]], "]"]]]]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["aa", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["aa", "3"], "]"]]]]], "\[Equal]", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", SubscriptBox["aa", "2"], ",", SubscriptBox["aa", "3"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", SubscriptBox["bb", "2"]]], "}"]], ",", "z"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["bb", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["bb", "2"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "1"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "1"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["aa", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["bb", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["bb", "2"], "+", "n"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["aa", "1"]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["aa", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["bb", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["bb", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "2"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "3"], "+", "n"]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "1"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "1"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "2"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "3"], "+", "n"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "2"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "2"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["aa", "2"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["bb", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["bb", "2"], "+", "n"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["aa", "2"]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["aa", "2"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["bb", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["bb", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "3"], "+", "n"]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "2"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "2"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "3"], "+", "n"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["bb", "1"], "-", SubscriptBox["aa", "3"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["bb", "2"], "-", SubscriptBox["aa", "3"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["aa", "3"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["bb", "1"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["bb", "2"], "+", "n"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["-", SubscriptBox["aa", "3"]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "n"]], "-", "1"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", SubscriptBox["aa", "3"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["bb", "1"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["bb", "2"], "+", "n"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "2"], "+", "n"]], ",", RowBox[List["2", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "1"], "+", "n"]]]], "}"]], ",", FractionBox["1", "z"]]], "]"]]]], RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "3"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "3"]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "2"], "+", "n"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", SubscriptBox["aa", "3"], "-", SubscriptBox["aa", "1"], "+", "n"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]]]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["aa", "1"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["aa", "2"], "]"]], " ", RowBox[List["Gamma", "[", SubscriptBox["aa", "3"], "]"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]], "&&", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "2"]]], "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["aa", "1"], "-", SubscriptBox["aa", "3"]]], "\[Element]", "Integers"]]]], "&&", RowBox[List["!", RowBox[List[RowBox[List[SubscriptBox["aa", "2"], "-", SubscriptBox["aa", "3"]]], "\[Element]", "Integers"]]]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02