Generalized hypergeometric function: Identities (formula 07.31.17.0012)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Identities

Functional identities

Relations between contiguous functions

http://functions.wolfram.com/07.31.17.0012.01

Input Form

HypergeometricPFQ[{a + 1, b + 1, Subscript[a, 3], \[Ellipsis], Subscript[a, p]}, {c + 1, d + 1, e + 1, Subscript[b, 4], \[Ellipsis], Subscript[b, q]}, z] - ((c d (a - e) (b - e))/(a b (c - e) (d - e))) HypergeometricPFQ[{a, b, Subscript[a, 3], \[Ellipsis], Subscript[a, p]}, {c, d, e + 1, Subscript[b, 4], \[Ellipsis], Subscript[b, q]}, z] - ((c e (a - d) (b - d))/(a b (c - d) (e - d))) HypergeometricPFQ[{a, b, Subscript[a, 3], \[Ellipsis], Subscript[a, p]}, {c, d + 1, e, Subscript[b, 4], \[Ellipsis], Subscript[b, q]}, z] - ((d e (a - c) (b - c))/(a b (d - c) (e - c))) HypergeometricPFQ[{a, b, Subscript[a, 3], \[Ellipsis], Subscript[a, p]}, {c + 1, d, e, Subscript[b, 4], \[Ellipsis], Subscript[b, q]}, z] == 0

Standard Form

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MathML Form

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</mo> <mrow> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> e </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <msub> <mi> b </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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[RowBox[List["a", "+", "1"]], ",", RowBox[List["b", "+", "1"]], ",", SubscriptBox["a", "3"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", RowBox[List["c", "+", "1"]]]], ",", RowBox[List["d", "+", "1"]], ",", RowBox[List["e", "+", "1"]], ",", SubscriptBox["b", "4"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> - </mo> <mtext> </mtext> <mrow> <mfrac> <mrow> <mi> c </mi> <mo> ⁢ </mo> <mi> d </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> e </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mi> p </mi> </msub> <msub> <mi> F </mi> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; 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</mo> <mi> c </mi> </mrow> <mo> , </mo> <mrow> <mi> d </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mi> e </mi> <mo> , </mo> <msub> <mi> b </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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["a", ",", "b", ",", SubscriptBox["a", "3"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", "c"]], ",", RowBox[List["d", "+", "1"]], ",", "e", ",", SubscriptBox["b", "4"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> d </mi> <mo> ⁢ </mo> <mi> e </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> e </mi> <mo> - </mo> <mi> c </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mi> p </mi> </msub> <msub> <mi> F </mi> <mi> q </mi> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <msub> <mi> a </mi> <mn> 3 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> ; </mo> <mrow> <mi> c </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> , </mo> <mi> d </mi> <mo> , </mo> <mi> e </mi> <mo> , </mo> <msub> <mi> b </mi> <mn> 4 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> ; </mo> <mi> z </mi> </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["a", ",", "b", ",", SubscriptBox["a", "3"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", "p"], ";", RowBox[List["c", "+", "1"]]]], ",", "d", ",", "e", ",", SubscriptBox["b", "4"], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["b", "q"], ";", "z"]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ⩵ </mo> <mn> 0 </mn> </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]