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http://functions.wolfram.com/07.32.03.0058.01
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HypergeometricPFQRegularized[{1, Subscript[a, 2], \[Ellipsis],
Subscript[a, q + 1]}, {Subscript[a, 2] + 1, \[Ellipsis],
Subscript[a, q + 1] + 1}, -1] ==
(1/(2 Product[Gamma[Subscript[a, j]], {j, 2, q + 1}]))
Sum[(PolyGamma[(Subscript[a, k] + 1)/2] - PolyGamma[Subscript[a, k]/2])
Product[If[l == k, 1, 1/(Subscript[a, l] - Subscript[a, k])],
{l, 2, q + 1}], {k, 2, q + 1}] /; Subscript[a, l] != Subscript[a, k] &&
2 <= l <= q + 1 && 2 <= k <= q + 1 && l != k
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "2"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", FractionBox[RowBox[List[SubscriptBox["a", "k"], "+", "1"]], "2"], "]"]], "-", RowBox[List["PolyGamma", "[", FractionBox[SubscriptBox["a", "k"], "2"], "]"]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", FractionBox["1", RowBox[List["(", RowBox[List[SubscriptBox["a", "l"], "-", SubscriptBox["a", "k"]]], ")"]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "l"], "\[NotEqual]", SubscriptBox["a", "k"]]], "\[And]", RowBox[List["2", "\[LessEqual]", "l", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["l", "\[NotEqual]", "k"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ; </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], ";", RowBox[List[SubscriptBox["a", "2"], "+", "1"]]]], ",", "\[Ellipsis]", ",", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]], ";", RowBox[List["-", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <msub> <mi> a </mi> <mi> k </mi> </msub> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <munder> <mrow> <mi> l </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> l </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> </munder> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mn> 1 </mn> <mrow> <msub> <mi> a </mi> <mi> l </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> l </mi> </msub> <mo> ≠ </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ∧ </mo> <mrow> <mn> 2 </mn> <mo> ≤ </mo> <mi> l </mi> <mo> ≤ </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mn> 2 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> l </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ; </mo> <mrow> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List["1", ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], ";", RowBox[List[SubscriptBox["a", "2"], "+", "1"]]]], ",", "\[Ellipsis]", ",", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]], ";", RowBox[List["-", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mfrac> <msub> <mi> a </mi> <mi> k </mi> </msub> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <munder> <mrow> <mi> l </mi> <mo> = </mo> <mn> 2 </mn> </mrow> <mrow> <mi> l </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> </munder> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mn> 1 </mn> <mrow> <msub> <mi> a </mi> <mi> l </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> l </mi> </msub> <mo> ≠ </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ∧ </mo> <mrow> <mn> 2 </mn> <mo> ≤ </mo> <mi> l </mi> <mo> ≤ </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mn> 2 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <mi> l </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a_", "2"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", RowBox[List["q_", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a_", "2"], "+", "1"]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", RowBox[List["q_", "+", "1"]]], "+", "1"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["a", "k"], "+", "1"]], ")"]]]], "]"]], "-", RowBox[List["PolyGamma", "[", FractionBox[SubscriptBox["a", "k"], "2"], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", FractionBox["1", RowBox[List[SubscriptBox["aa", "l"], "-", SubscriptBox["a", "k"]]]]]], "]"]]]]]]]], RowBox[List["2", " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "2"]], RowBox[List["q", "+", "1"]]], RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["aa", "l"], "\[NotEqual]", SubscriptBox["a", "k"]]], "&&", RowBox[List["2", "\[LessEqual]", "l", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["2", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "&&", RowBox[List["l", "\[NotEqual]", "k"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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