Generalized hypergeometric function: Transformations (formula 07.31.16.0001)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]

Transformations

Products, sums, and powers of the direct function

Products of the direct function

http://functions.wolfram.com/07.31.16.0001.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, c z] HypergeometricPFQ[{Subscript[\[Alpha], 1], \[Ellipsis], Subscript[\[Alpha], r]}, {Subscript[\[Beta], 1], \[Ellipsis], Subscript[\[Beta], s]}, d z] == Sum[Subscript[c, k] z^k, {k, 0, Infinity}] /; Subscript[c, k] == ((d^k Product[Pochhammer[Subscript[\[Alpha], j], k], {j, 1, r}])/(k! Product[Pochhammer[Subscript[\[Beta], j], k], {j, 1, s}])) HypergeometricPFQ[{-k, 1 - Subscript[\[Beta], 1] - k, \[Ellipsis], 1 - Subscript[\[Beta], s] - k, Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {1 - Subscript[\[Alpha], 1] - k, \[Ellipsis], 1 - Subscript[\[Alpha], r] - k, Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, ((-1)^(r + s - 1) c)/d] || Subscript[c, k] == ((c^k Product[Pochhammer[Subscript[a, j], k], {j, 1, p}])/(k! Product[Pochhammer[Subscript[b, j], k], {j, 1, q}])) HypergeometricPFQ[{-k, 1 - Subscript[b, 1] - k, \[Ellipsis], 1 - Subscript[b, q] - k, Subscript[\[Alpha], 1], \[Ellipsis], Subscript[\[Alpha], r]}, {1 - Subscript[a, 1] - k, \[Ellipsis], 1 - Subscript[a, p] - k, Subscript[\[Beta], 1], \[Ellipsis], Subscript[\[Beta], s]}, ((-1)^(p + q - 1) d)/c]

Standard Form

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MathML Form

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</mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["p", TraditionalForm]], SubscriptBox["F", FormBox["q", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", SubscriptBox["b", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", RowBox[List["c", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mi> r </mi> </msub> <msub> <mi> F </mi> <mi> s </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> α </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> α </mi> <mi> r </mi> </msub> <mo> ; </mo> <msub> <mi> β </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> β </mi> <mi> s </mi> </msub> <mo> ; </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["r", TraditionalForm]], SubscriptBox["F", FormBox["s", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Alpha]", "r"], ";", SubscriptBox["\[Beta]", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Beta]", "s"], ";", RowBox[List["d", " ", "z"]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> d </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> r </mi> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> α </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Alpha]", "j"], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> s </mi> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> β </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["\[Beta]", "j"], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> p </mi> <mo> + </mo> <mi> s </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mi> q </mi> <mo> + </mo> <mi> r </mi> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> β </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> β </mi> <mi> s </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> α </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> α </mi> <mi> r </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> r </mi> <mo> + </mo> <mi> s </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> c </mi> </mrow> <mi> d </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["p", "+", "s", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["q", "+", "r"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List["1", "-", SubscriptBox["\[Beta]", "1"], "-", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["\[Beta]", "s"], "-", "k"]], ",", SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "1"], "-", "k"]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["\[Alpha]", "r"], "-", "k"]], ",", SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["r", "+", "s", "-", "1"]]], "c"]], "d"]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ∨ </mo> <mrow> <msub> <mi> c </mi> <mi> k </mi> </msub> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> c </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "j"], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["b", "j"], ")"]], "k"], Pochhammer] </annotation> </semantics> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> q </mi> <mo> + </mo> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mi> p </mi> <mo> + </mo> <mi> s </mi> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> α </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> α </mi> <mi> r </mi> </msub> <mo> ; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> r </mi> </msub> <mo> - </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> β </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> β </mi> <mi> s </mi> </msub> <mo> ; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mi> c </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["q", "+", "r", "+", "1"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["p", "+", "s"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[RowBox[List["-", "k"]], ",", RowBox[List["1", "-", SubscriptBox["b", "1"], "-", "k"]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["b", "q"], "-", "k"]], ",", SubscriptBox["\[Alpha]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Alpha]", "r"], ";", RowBox[List["1", "-", SubscriptBox["a", "1"], "-", "k"]]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "-", SubscriptBox["a", "r"], "-", "k"]], ",", SubscriptBox["\[Beta]", "1"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["\[Beta]", "s"], ";", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "+", "q", "-", "1"]]], "d"]], "c"]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

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

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]