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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.03.0027.01

 Input Form

 HypergeometricPFQ[{1, a, a, Subscript[a, 4], \[Ellipsis], Subscript[a, q + 1]}, {a + 1, a + 1, Subscript[a, 4] + 1, \[Ellipsis], Subscript[a, q + 1] + 1}, 1] == (-a^2) Product[Subscript[a, j], {j, 4, q + 1}] (Sum[Product[If[l == k, 1, PolyGamma[Subscript[a, k]]/ ((Subscript[a, l] - Subscript[a, k]) (a - Subscript[a, k])^2)], {l, 4, q + 1}] + (PolyGamma[a]/(a - Subscript[a, k])) Product[1/(Subscript[a, l] - a), {l, 4, q + 1}], {k, 4, q + 1}] - PolyGamma[1, a] Product[1/(Subscript[a, l] - a), {l, 4, q + 1}]) /; Subscript[a, l] != Subscript[a, k] && 4 <= l <= q + 1 && 4 <= k <= q + 1 && l != k

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "a", ",", "a", ",", SubscriptBox["a", "4"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["q", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["a", "+", "1"]], ",", RowBox[List["a", "+", "1"]], ",", RowBox[List[SubscriptBox["a", "4"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["a", "2"]]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "4"]], RowBox[List["q", "+", "1"]]], SubscriptBox["a", "j"]]], ")"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], RowBox[List["If", "[", RowBox[List[RowBox[List["l", "\[Equal]", "k"]], ",", "1", ",", FractionBox[RowBox[List["PolyGamma", "[", SubscriptBox["a", "k"], "]"]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["a", "l"], "-", SubscriptBox["a", "k"]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["a", "k"]]], ")"]], "2"]]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["PolyGamma", "[", "a", "]"]], RowBox[List["a", "-", SubscriptBox["a", "k"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], FractionBox["1", RowBox[List[SubscriptBox["a", "l"], "-", "a"]]]]]]]]], ")"]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "a"]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "4"]], RowBox[List["q", "+", "1"]]], FractionBox["1", RowBox[List[SubscriptBox["a", "l"], "-", "a"]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "l"], "\[NotEqual]", SubscriptBox["a", "k"]]], "\[And]", RowBox[List["4", "\[LessEqual]", "l", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["4", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["q", "+", "1"]]]], "\[And]", RowBox[List["l", "\[NotEqual]", "k"]]]]]]]]

 MathML Form

 q + 1 F q ( 1 , a , a , a 4 , , a q + 1 ; a + 1 , a + 1 , a 4 + 1 , , a q + 1 + 1 ; 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["a", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[SubscriptBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", "4"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] - a 2 ( j = 4 q + 1 a j ) ( k = 4 q + 1 ( ψ TagBox["\[Psi]", PolyGamma] ( a ) a - a k l = 4 q + 1 1 a l - a + l = 4 l k q + 1 ψ TagBox["\[Psi]", PolyGamma] ( a k ) ( a l - a k ) ( a - a k ) 2 ) - ψ TagBox["\[Psi]", PolyGamma] ( 1 ) ( a ) l = 4 q + 1 1 a l - a ) /; a l a k 4 l q + 1 4 k q + 1 l k FormBox RowBox RowBox TagBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox F FormBox q TraditionalForm RowBox ( RowBox TagBox 1 HypergeometricPFQ Rule Editable , TagBox a HypergeometricPFQ Rule Editable , TagBox a HypergeometricPFQ Rule Editable , TagBox SubscriptBox a 4 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , RowBox TagBox SubscriptBox a RowBox q + 1 HypergeometricPFQ Rule Editable ; TagBox TagBox RowBox TagBox RowBox a + 1 HypergeometricPFQ Rule Editable , TagBox RowBox a + 1 HypergeometricPFQ Rule Editable , TagBox RowBox SubscriptBox a 4 + 1 HypergeometricPFQ Rule Editable , TagBox HypergeometricPFQ Rule Editable , TagBox RowBox SubscriptBox a RowBox q + 1 + 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox 1 HypergeometricPFQ Rule Editable ) InterpretTemplate Function HypergeometricPFQ Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQ RowBox RowBox - SuperscriptBox a 2 RowBox ( RowBox UnderoverscriptBox RowBox j = 4 RowBox q + 1 SubscriptBox a j ) RowBox ( RowBox RowBox UnderoverscriptBox RowBox k = 4 RowBox q + 1 ErrorBox RowBox ( RowBox RowBox FractionBox RowBox TagBox ψ PolyGamma ( a ) RowBox a - SubscriptBox a k RowBox UnderoverscriptBox RowBox l = 4 RowBox q + 1 FractionBox 1 RowBox SubscriptBox a l - a + RowBox UnderoverscriptBox UnderscriptBox RowBox l = 4 RowBox l k RowBox q + 1 FractionBox RowBox TagBox ψ PolyGamma ( SubscriptBox a k ) RowBox RowBox ( RowBox SubscriptBox a l - SubscriptBox a k ) SuperscriptBox RowBox ( RowBox a - SubscriptBox a k ) 2 ) - RowBox RowBox SuperscriptBox TagBox ψ PolyGamma RowBox ( 1 ) ( a ) RowBox UnderoverscriptBox RowBox l = 4 RowBox q + 1 FractionBox 1 RowBox SubscriptBox a l - a ) /; RowBox RowBox SubscriptBox a l SubscriptBox a k RowBox 4 l RowBox q + 1 RowBox 4 k RowBox q + 1 RowBox l k TraditionalForm [/itex]

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

 2001-10-29