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 HypergeometricPFQ

 http://functions.wolfram.com/07.31.17.0023.01

 Input Form

 Sum[(Product[If[j == k, 1, Gamma[Subscript[b, j] - Subscript[b, k]]] Product[Gamma[1 + Subscript[b, k] - Subscript[a, j]], {j, 1, n}], {j, 1, m}]/Product[Gamma[Subscript[a, j] - Subscript[b, k]] Product[Gamma[1 - Subscript[b, j] + Subscript[b, k]], {j, m + 1, q + 1}], {j, n + 1, q + 1}]) z^Subscript[b, k] HypergeometricPFQ[{1 + Subscript[b, k] - Subscript[a, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[a, q + 1]}, {1 + Subscript[b, k] - Subscript[b, 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, k - 1], 1 + Subscript[b, k] - Subscript[b, k + 1], \[Ellipsis], 1 + Subscript[b, k] - Subscript[b, q + 1]}, (-1)^(q - m - n + 1) z], {k, 1, m}] == Sum[(Product[If[j == k, 1, Gamma[Subscript[a, k] - Subscript[a, j]]] Product[Gamma[1 + Subscript[b, j] - Subscript[a, k]], {j, 1, m}], {j, 1, n}]/Product[Gamma[Subscript[a, k] - Subscript[b, j]] Product[Gamma[1 + Subscript[a, j] - Subscript[a, k]], {j, n + 1, q + 1}], {j, m + 1, q + 1}]) z^(Subscript[a, k] - 1) HypergeometricPFQ[{1 + Subscript[b, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[b, q + 1] - Subscript[a, k]}, {1 + Subscript[a, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, k - 1] - Subscript[a, k], 1 + Subscript[a, k + 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, q + 1] - Subscript[a, k]}, (-1)^(q - m - n + 1)/z], {k, 1, n}] /; Element[m, Integers] && m > 0 && Element[n, Integers] && n > 0 && Element[q, Integers] && q >= 0 && m <= q + 1 && n <= q + 1 && (m + n - q > 2 || (m + n - q == 2 && !IntervalMemberQ[Interval[{-1, 0}], z]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "m"], RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "m"], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["j", "\[Equal]", "k"]], ",", "1", ",", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["b", "j"], "-", SubscriptBox["b", "k"]]], "]"]]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List["Gamma", "[", RowBox[List["1", "+", SubscriptBox["b", "k"], "-", SubscriptBox["a", "j"]]], "]"]]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["n", "+", "1"]]]], RowBox[List["q", "+", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "j"], "-", SubscriptBox["b", "k"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["m", "+", "1"]]]], RowBox[List["q", "+", "1"]]], 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 MathML Form

 k = 1 m j = 1 j k m Γ ( b j - b k ) j = 1 n Γ ( 1 - a j + b k ) j = n + 1 q + 1 Γ ( a j - b k ) j = m + 1 q + 1 Γ ( 1 - b j + b k ) z b k q + 1 F q ( 1 - a 1 + b k , , 1 - a q + 1 + b k ; 1 - b 1 + b k , , 1 - b k - 1 + b k , 1 - b k + 1 + b k , , 1 - b q + 1 + b k ; ( - 1 ) q - m - n + 1 z ) 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[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["b", "1"], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "-", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["k", "+", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[TagBox[RowBox[List["1", "-", SubscriptBox["b", RowBox[List["q", "+", "1"]]], "+", SubscriptBox["b", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n", "+", "1"]]], " ", "z"]], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] k = 1 n j = 1 j k n Γ ( a k - a j ) j = 1 m Γ ( 1 - a k + b j ) j = m + 1 q + 1 Γ ( a k - b j ) j = n + 1 q + 1 Γ ( a j - a k + 1 ) z a k - 1 q + 1 F q ( 1 - a k + b 1 , , 1 - a k + b q + 1 ; 1 + a 1 - a k , , 1 + a k - 1 - a k , 1 + a k + 1 - a k , , 1 + a q + 1 - a k ; ( - 1 ) q - m - n + 1 z ) 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[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", RowBox[List["q", "+", "1"]]]]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "+", SubscriptBox["a", "1"], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "-", "1"]]], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "+", SubscriptBox["a", RowBox[List["k", "+", "1"]]], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQRegularized, Rule[Editable, True]], ",", TagBox[TagBox[RowBox[List["1", "+", SubscriptBox["a", RowBox[List["q", "+", "1"]]], "-", SubscriptBox["a", "k"]]], HypergeometricPFQRegularized, Rule[Editable, True]], HypergeometricPFQRegularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]], ";", TagBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n", "+", "1"]]], "z"], HypergeometricPFQRegularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQRegularized] /; m + n + q m q + 1 n q + 1 ( m + n - q > 2 ( m + n - q 2 z ( - 1 , 0 ) ) ) FormBox RowBox RowBox RowBox UnderoverscriptBox RowBox k = 1 m ErrorBox RowBox FractionBox RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k m RowBox Γ ( RowBox SubscriptBox b j - SubscriptBox b k ) RowBox UnderoverscriptBox RowBox j = 1 n RowBox Γ ( RowBox 1 - SubscriptBox a j + SubscriptBox b k ) RowBox UnderoverscriptBox RowBox j = RowBox n + 1 RowBox q + 1 RowBox RowBox Γ ( RowBox SubscriptBox a j - SubscriptBox b k ) RowBox UnderoverscriptBox RowBox j = RowBox m + 1 RowBox q + 1 RowBox Γ ( RowBox 1 - SubscriptBox b j + SubscriptBox b k ) SuperscriptBox z SubscriptBox b k TagBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox F FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox 1 - SubscriptBox a 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - SubscriptBox a RowBox q + 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox TagBox RowBox TagBox RowBox 1 - SubscriptBox b 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k - 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - SubscriptBox b RowBox k + 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox TagBox RowBox 1 - SubscriptBox b RowBox q + 1 + SubscriptBox b k HypergeometricPFQRegularized Rule Editable HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox q - m - n + 1 z HypergeometricPFQRegularized Rule Editable ) InterpretTemplate Function HypergeometricPFQRegularized Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQRegularized RowBox UnderoverscriptBox RowBox k = 1 n RowBox FractionBox RowBox UnderoverscriptBox UnderscriptBox RowBox j = 1 RowBox j k n RowBox Γ ( RowBox SubscriptBox a k - SubscriptBox a j ) RowBox UnderoverscriptBox RowBox j = 1 m RowBox Γ ( RowBox 1 - SubscriptBox a k + SubscriptBox b j ) RowBox UnderoverscriptBox RowBox j = RowBox m + 1 RowBox q + 1 RowBox RowBox Γ ( RowBox SubscriptBox a k - SubscriptBox b j ) RowBox UnderoverscriptBox RowBox j = RowBox n + 1 RowBox q + 1 RowBox Γ ( RowBox SubscriptBox a j - SubscriptBox a k + 1 ) SuperscriptBox z RowBox SubscriptBox a k - 1 TagBox TagBox RowBox RowBox SubscriptBox FormBox RowBox q + 1 TraditionalForm SubscriptBox F FormBox q TraditionalForm RowBox ( RowBox TagBox TagBox RowBox TagBox RowBox 1 - SubscriptBox a k + SubscriptBox b 1 HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 - SubscriptBox a k + SubscriptBox b RowBox q + 1 HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox TagBox RowBox TagBox RowBox 1 + SubscriptBox a 1 - SubscriptBox a k HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 + SubscriptBox a RowBox k - 1 - SubscriptBox a k HypergeometricPFQRegularized Rule Editable , TagBox RowBox 1 + SubscriptBox a RowBox k + 1 - SubscriptBox a k HypergeometricPFQRegularized Rule Editable , TagBox HypergeometricPFQRegularized Rule Editable , TagBox TagBox RowBox 1 + SubscriptBox a RowBox q + 1 - SubscriptBox a k HypergeometricPFQRegularized Rule Editable HypergeometricPFQRegularized Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQRegularized Rule Editable ; TagBox FractionBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox q - m - n + 1 z HypergeometricPFQRegularized Rule Editable ) InterpretTemplate Function HypergeometricPFQRegularized Slot 1 Slot 2 Slot 3 Rule Editable HypergeometricPFQRegularized /; RowBox RowBox m SuperscriptBox + RowBox n SuperscriptBox + RowBox q RowBox m RowBox q + 1 RowBox n RowBox q + 1 RowBox ( RowBox RowBox RowBox m + n - q > 2 RowBox ( RowBox RowBox RowBox m + n - q 2 RowBox z RowBox ( RowBox RowBox - 1 , 0 ) ) ) TraditionalForm [/itex]

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

 2001-10-29