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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.20.0003.01

 Input Form

 Derivative[{0, \[Ellipsis], 0}, {1, 0, \[Ellipsis], 0}, 0][HypergeometricPFQ][ {Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == PolyGamma[Subscript[b, 1]] HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] - Sum[(PolyGamma[Subscript[b, 1] + k] Product[Pochhammer[Subscript[a, j], k], {j, 1, p}] z^k)/(k! Product[Pochhammer[Subscript[b, j], k], {j, 1, q}]), {k, 0, Infinity}] /; (q == p - 1 && Abs[z] < 1) || q >= p

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List[RowBox[List["{", RowBox[List["0", ",", "\[Ellipsis]", ",", "0"]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "0", ",", "\[Ellipsis]", ",", "0"]], "}"]], ",", "0"]], "]"]], "[", "HypergeometricPFQ", "]"]], "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["b", "1"], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[SubscriptBox["b", "1"], "+", "k"]], "]"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "p"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "j"], ",", "k"]], "]"]]]], " ", ")"]], SuperscriptBox["z", "k"]]]]], RowBox[List[RowBox[List["k", "!"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "q"], " ", RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "j"], ",", "k"]], "]"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["q", "\[Equal]", RowBox[List["p", "-", "1"]]]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]], "\[Or]", RowBox[List["q", "\[GreaterEqual]", "p"]]]]]]]]

 MathML Form

 p F q ( { 0 , , 0 } , { 1 , 0 , , 0 } , 0 ) ( a 1 , , a p TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; b 1 , , b q TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) ψ TagBox["\[Psi]", PolyGamma] ( b 1 ) p F q ( a 1 , , a p TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; b 1 TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]] , TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]] , b q TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) - k = 0 ψ TagBox["\[Psi]", PolyGamma] ( k + b 1 ) ( j = 1 p ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] ) z k k ! j = 1 q ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] /; q p - 1 "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 q p p F q ( { 0 , , 0 } , { 1 , 0 , , 0 } , 0 ) ( a 1 , , a p TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; b 1 , , b q TagBox[TagBox[RowBox[List[TagBox[RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) ψ TagBox["\[Psi]", PolyGamma] ( b 1 ) p F q ( a 1 , , a p TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; b 1 TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]] , TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]] , b q TagBox[SubscriptBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) - k = 0 ψ TagBox["\[Psi]", PolyGamma] ( k + b 1 ) ( j = 1 p ( a j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] ) z k k ! j = 1 q ( b j ) k TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] /; q p - 1 "\[LeftBracketingBar]" z "\[RightBracketingBar]" < 1 q p [/itex]

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

 2001-10-29