Generalized hypergeometric function: Specific values (formula 07.31.03.0041)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Specific values

Specialized values

Case _q+1F_qat z==1

http://functions.wolfram.com/07.31.03.0041.01

Input Form

HypergeometricPFQ[{-n, a/(q + 1), (a + 1)/(q + 1), \[Ellipsis], (a + q)/(q + 1), b/q, (b + 1)/q, \[Ellipsis], (b + q - 1)/q}, {a/q, (a + 1)/q, \[Ellipsis], (a + q - 1)/q, (1 + b - n)/(q + 1), (2 + b - n)/(q + 1), \[Ellipsis], (q + 1 + b - n)/(q + 1)}, 1] == -(Pochhammer[a - b, n]/((b + n q) Pochhammer[1 - b, n - 1])) /; Element[n, Integers] && n > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", "n"]], ",", FractionBox["a", RowBox[List["q", "+", "1"]]], ",", FractionBox[RowBox[List["a", "+", "1"]], RowBox[List["q", "+", "1"]]], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["a", "+", "q"]], RowBox[List["q", "+", "1"]]], ",", FractionBox["b", "q"], ",", FractionBox[RowBox[List["b", "+", "1"]], "q"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["b", "+", "q", "-", "1"]], "q"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["a", "q"], ",", FractionBox[RowBox[List["a", "+", "1"]], "q"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["a", "+", "q", "-", "1"]], "q"], ",", FractionBox[RowBox[List["1", "+", "b", "-", "n"]], RowBox[List["q", "+", "1"]]], ",", FractionBox[RowBox[List["2", "+", "b", "-", "n"]], RowBox[List["q", "+", "1"]]], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["q", "+", "1", "+", "b", "-", "n"]], RowBox[List["q", "+", "1"]]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "-", "b"]], ",", "n"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["n", " ", "q"]]]], ")"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", RowBox[List["n", "-", "1"]]]], "]"]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> , </mo> <mfrac> <mi> a </mi> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> q </mi> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> , </mo> <mfrac> <mi> b </mi> <mi> q </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> b </mi> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> <mo> ; </mo> <mrow> <mfrac> <mi> a </mi> <mi> q </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <mo> ; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["2", "q"]], "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List[RowBox[List["2", "q"]], "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List["-", "n"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["a", RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "1"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "q"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["b", "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "+", "1"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["b", "+", "q", "-", "1"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["a", "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "1"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "q", "-", "1"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "-", "n", "+", "1"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "-", "n", "+", "2"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "-", "n", "+", "q", "+", "1"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["1", HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mfrac> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], "n"], Pochhammer] </annotation> </semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], RowBox[List["n", "-", "1"]]], Pochhammer] </annotation> </semantics> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <ci> n </ci> <ci> q </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </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]