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 HypergeometricPFQRegularized

 http://functions.wolfram.com/07.32.03.0010.01

 Input Form

 HypergeometricPFQRegularized[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[a, 1] + Subscript[m, 1], \[Ellipsis], Subscript[a, n] + Subscript[m, n], Subscript[b, n + 1], \[Ellipsis], Subscript[b, q]}, z] == (1/Product[Gamma[Subscript[a, j]] (Subscript[m, j] - 1)!, {j, 1, n}]) Sum[\[Ellipsis] Sum[Gamma[Subscript[a, k] + Subscript[j, k]] Product[(Pochhammer[1 - Subscript[m, l], Subscript[j, l]]/ Subscript[j, l]!) Product[If[i == k, 1, 1/(Subscript[a, i] + Subscript[j, i] - Subscript[a, k] - Subscript[j, k])] HypergeometricPFQRegularized[{Subscript[a, k] + Subscript[j, k], Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}, {Subscript[a, k] + Subscript[j, k] + 1, Subscript[b, n + 1], \[Ellipsis], Subscript[b, q]}, z], {i, 1, n}], {l, 1, n}], {Subscript[j, n], 0, Subscript[m, n] - 1}], {k, 1, n}, {Subscript[j, 1], 0, Subscript[m, 1] - 1}] /; Element[Subscript[m, n], Integers] && Subscript[m, n] > 0 && n <= q

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["m", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "n"], "+", SubscriptBox["m", "n"]]], ",", SubscriptBox["b", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]], RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["m", "j"], "-", "1"]], ")"]], "!"]]]], ")"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", "0"]], RowBox[List[SubscriptBox["m", "1"], "-", "1"]]], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "n"], "=", "0"]], RowBox[List[SubscriptBox["m", "n"], "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"]]], "]"]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "1"]], "n"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["m", "l"]]], ",", SubscriptBox["j", "l"]]], "]"]], RowBox[List[SubscriptBox["j", "l"], "!"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], "n"], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["i", "\[Equal]", "k"]], ",", "1", ",", FractionBox["1", RowBox[List[SubscriptBox["a", "i"], "+", SubscriptBox["j", "i"], "-", SubscriptBox["a", "k"], "-", SubscriptBox["j", "k"]]]]]], "]"]], RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"]]], ",", SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"], "+", "1"]], ",", SubscriptBox["b", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]], ",", "z"]], "]"]]]]]]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["m", "n"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["m", "n"], ">", "0"]], "\[And]", RowBox[List["n", "\[LessEqual]", "q"]]]]]]]]

 MathML Form

 p F ~ q ( a 1 , , a p TagBox[TagBox[TagBox[TagBox[RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; a 1 + m 1 , , a n + m n , b n + 1 , , b q TagBox[TagBox[TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["a", "1"], "+", SubscriptBox["m", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "n"], "+", SubscriptBox["m", "n"]]], ",", SubscriptBox["b", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) 1 j = 1 n Γ ( a j ) ( m j - 1 ) ! k = 1 n j 1 = 0 m 1 - 1 j n = 0 m n - 1 Γ ( a k + j k ) l = 1 n ( 1 - m l ) j l TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["m", "l"]]], ")"]], SubscriptBox["j", "l"]], Pochhammer] j l ! i = 1 i k n 1 a i + j i - a k - j k p - n + 1 F ~ q - n + 1 ( a k + j k , a n + 1 , , a p TagBox[TagBox[TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"]]], ",", SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; a k + j k + 1 , b n + 1 , , b q TagBox[TagBox[TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"], "+", "1"]], ",", SubscriptBox["b", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, True]] ; z TagBox["z", HypergeometricPFQ, Rule[Editable, True]] ) FormBox RowBox RowBox RowBox SubscriptBox ErrorBox FormBox p TraditionalForm SubscriptBox OverscriptBox F ~ q RowBox ( RowBox TagBox TagBox TagBox TagBox RowBox SubscriptBox a 1 , , SubscriptBox a p InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox TagBox TagBox RowBox RowBox SubscriptBox a 1 + SubscriptBox m 1 , , RowBox SubscriptBox a n + SubscriptBox m n , SubscriptBox b RowBox n + 1 , , SubscriptBox b q InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) RowBox FractionBox 1 RowBox UnderoverscriptBox RowBox j = 1 n RowBox RowBox Γ ( SubscriptBox a j ) RowBox RowBox ( RowBox SubscriptBox m j - 1 ) ! RowBox UnderoverscriptBox RowBox k = 1 n RowBox UnderoverscriptBox RowBox SubscriptBox j 1 = 0 RowBox SubscriptBox m 1 - 1 RowBox RowBox UnderoverscriptBox RowBox SubscriptBox j n = 0 RowBox SubscriptBox m n - 1 RowBox RowBox Γ ( RowBox SubscriptBox a k + SubscriptBox j k ) RowBox UnderoverscriptBox RowBox l = 1 n RowBox FractionBox TagBox SubscriptBox RowBox ( RowBox 1 - SubscriptBox m l ) SubscriptBox j l Pochhammer RowBox SubscriptBox j l ! RowBox UnderoverscriptBox UnderscriptBox RowBox i = 1 RowBox i k n RowBox FractionBox 1 RowBox SubscriptBox a i + SubscriptBox j i - SubscriptBox a k - SubscriptBox j k RowBox RowBox SubscriptBox FormBox RowBox p - n + 1 TraditionalForm SubscriptBox OverscriptBox F ~ RowBox q - n + 1 RowBox ( RowBox TagBox TagBox TagBox TagBox RowBox RowBox SubscriptBox a k + SubscriptBox j k , SubscriptBox a RowBox n + 1 , , SubscriptBox a p InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox TagBox TagBox TagBox RowBox RowBox SubscriptBox a k + SubscriptBox j k + 1 , SubscriptBox b RowBox n + 1 , , SubscriptBox b q InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable InterpretTemplate Function SlotSequence 1 HypergeometricPFQ Rule Editable ; TagBox z HypergeometricPFQ Rule Editable ) TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a_", "1"], "+", SubscriptBox["m_", "1"]]], ",", "\[Ellipsis]_", ",", RowBox[List[SubscriptBox["a_", "n_"], "+", SubscriptBox["m_", "n_"]]], ",", SubscriptBox["b_", RowBox[List["n_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", "0"]], RowBox[List[SubscriptBox["mm", "1"], "-", "1"]]], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "n"], "=", "0"]], RowBox[List[SubscriptBox["mm", "n"], "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["l", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", SubscriptBox["mm", "l"]]], ",", SubscriptBox["j", "l"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], "n"], RowBox[List[RowBox[List["If", "[", RowBox[List[RowBox[List["i", "\[Equal]", "k"]], ",", "1", ",", FractionBox["1", RowBox[List[SubscriptBox["aa", "i"], "+", SubscriptBox["j", "i"], "-", SubscriptBox["a", "k"], "-", SubscriptBox["j", "k"]]]]]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"]]], ",", SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["aa", "p"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "k"], "+", SubscriptBox["j", "k"], "+", "1"]], ",", SubscriptBox["b", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["bb", "q"]]], "}"]], ",", "z"]], "]"]]]]]]]], RowBox[List[SubscriptBox["j", "l"], "!"]]]]]]]]]]]]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[RowBox[List["Gamma", "[", SubscriptBox["a", "j"], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[SubscriptBox["m", "j"], "-", "1"]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["mm", "n"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["mm", "n"], ">", "0"]], "&&", RowBox[List["n", "\[LessEqual]", "q"]]]]]]]]]]

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

 2001-10-29