Generalized hypergeometric function: Identities (formula 07.31.17.0020)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Identities

Functional identities

Division on even and odd parts and generalization

http://functions.wolfram.com/07.31.17.0020.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], \[Ellipsis], Subscript[a, p]}, {Subscript[b, 1], \[Ellipsis], Subscript[b, q]}, z] == SuperPlus[A][z] + SuperMinus[A][z] /; SuperPlus[A][z] == HypergeometricPFQ[{Subscript[a, 1]/2, \[Ellipsis], Subscript[a, p]/2, (Subscript[a, 1] + 1)/2, \[Ellipsis], (Subscript[a, p] + 1)/2}, {1/2, Subscript[b, 1]/2, \[Ellipsis], Subscript[b, q]/2, (Subscript[b, 1] + 1)/2, \[Ellipsis], (Subscript[b, q] + 1)/2}, 4^(p - q - 1) z^2] && SuperMinus[A][z] == ((z Product[Subscript[a, j], {j, 1, p}])/ Product[Subscript[b, j], {j, 1, q}]) HypergeometricPFQ[ {(Subscript[a, 1] + 1)/2, \[Ellipsis], (Subscript[a, p] + 1)/2, (Subscript[a, 1] + 2)/2, \[Ellipsis], (Subscript[a, p] + 2)/2}, {3/2, (Subscript[b, 1] + 1)/2, \[Ellipsis], (Subscript[b, q] + 1)/2, (Subscript[b, 1] + 2)/2, \[Ellipsis], (Subscript[b, q] + 2)/2}, 4^(p - q - 1) z^2]

Standard Form

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MathML Form

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</mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <msup> <mn> 4 </mn> <mrow> <mi> p </mi> <mo> - </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["2", "p"]], TraditionalForm]], SubscriptBox["F", RowBox[List["2", FormBox[RowBox[List["q", "+", "1"]], TraditionalForm]]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], "2"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "1"]], "2"], ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "2"]], "2"], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List[SubscriptBox["a", "p"], "+", "2"]], "2"], ";", FractionBox["3", "2"]]], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "1"]], "2"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "1"]], "2"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "2"]], "2"], ",", "\[Ellipsis]", ",", RowBox[List[FractionBox[RowBox[List[SubscriptBox["b", "q"], "+", "2"]], "2"], ";", RowBox[List[SuperscriptBox["4", RowBox[List["p", "-", "q", "-", "1"]]], SuperscriptBox["z", "2"]]]]]]], ")"]]]], 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]