|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.31.03.0040.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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 + 1)/q, (a + 2)/q, \[Ellipsis], (a + q)/q, (b - n)/(q + 1),
(b - n + 1)/(q + 1), \[Ellipsis], (b - n + q)/(q + 1)}, 1] ==
Pochhammer[1 + a - b, n]/Pochhammer[1 - b, n] /;
Element[n, Integers] && n > 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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[RowBox[List["a", "+", "1"]], "q"], ",", FractionBox[RowBox[List["a", "+", "2"]], "q"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["a", "+", "q"]], "q"], ",", FractionBox[RowBox[List["b", "-", "n"]], RowBox[List["q", "+", "1"]]], ",", FractionBox[RowBox[List["b", "-", "n", "+", "1"]], RowBox[List["q", "+", "1"]]], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["b", "-", "n", "+", "q"]], RowBox[List["q", "+", "1"]]]]], "}"]], ",", "1"]], "]"]], "\[Equal]", FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "-", "b"]], ",", "n"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "n"]], "]"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<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> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mi> q </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> q </mi> </mrow> <mi> q </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mi> n </mi> </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> 1 </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> </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[RowBox[List["a", "+", "1"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "2"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "q"]], "q"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "-", "n"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "-", "n", "+", "1"]], RowBox[List["q", "+", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["b", "-", "n", "+", "q"]], 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> <mfrac> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["a", "-", "b", "+", "1"]], ")"]], "n"], Pochhammer] </annotation> </semantics> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], "n"], Pochhammer] </annotation> </semantics> </mfrac> </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 /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </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)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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] | |
|
|
|