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http://functions.wolfram.com/07.26.17.0022.01
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HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2]},
{Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] ==
Sum[(z^k/k!) ((Pochhammer[Subscript[a, 1], k] Pochhammer[Subscript[a, 2],
k])/Product[Pochhammer[Subscript[b, j], k], {j, 1, 3}])
HypergeometricPFQ[{1, (Subscript[a, 1] + k)/n, \[Ellipsis],
(Subscript[a, 1] + k + n - 1)/n, (Subscript[a, 2] + k)/n, \[Ellipsis],
(Subscript[a, 2] + k + n - 1)/n}, {(k + 1)/n, \[Ellipsis], (k + n)/n,
(Subscript[b, 1] + k)/n, \[Ellipsis], (Subscript[b, 1] + k + n - 1)/n,
(Subscript[b, 2] + k)/n, \[Ellipsis], (Subscript[b, 2] + k + n - 1)/n,
(Subscript[b, 3] + k)/n, \[Ellipsis], (Subscript[b, 3] + k + n - 1)/n},
z^n/n^(2 n)], {k, 0, n - 1}]
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Cell[BoxData[RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", SubscriptBox["b", "2"], ",", SubscriptBox["b", "3"]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[SuperscriptBox["z", "k"], RowBox[List["k", "!"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "1"], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["a", "2"], ",", "k"]], "]"]]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], "3"], RowBox[List["Pochhammer", "[", RowBox[List[SubscriptBox["b", "j"], ",", "k"]], "]"]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["a", "2"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["a", "2"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["k", "+", "1"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List["k", "+", "n"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "1"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "2"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "2"], "+", "k", "+", "n", "-", "1"]], "n"], ",", FractionBox[RowBox[List[SubscriptBox["b", "3"], "+", "k"]], "n"], ",", "\[Ellipsis]", ",", FractionBox[RowBox[List[SubscriptBox["b", "3"], "+", "k", "+", "n", "-", "1"]], "n"]]], "}"]], ",", RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", "2"]], "n"]]], SuperscriptBox["z", "n"]]]]], "]"]]]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> ; </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["3", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["a", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[SubscriptBox["b", "1"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "3"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "1"], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", SubscriptBox["a", "2"], ")"]], "k"], Pochhammer] </annotation> </semantics> </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> <mn> 3 </mn> </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> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <msub> <mi> b </mi> <mn> 2 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <msub> <mi> b </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <mrow> <msup> <mi> n </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List[RowBox[List["2", "n"]], "+", "1"]], TraditionalForm]], SubscriptBox["F", RowBox[List["4", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "1"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["a", "1"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["a", "2"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["a", "2"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[FractionBox[RowBox[List["k", "+", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]]]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "1"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "1"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "2"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "2"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["k", "+", SubscriptBox["b", "3"]]], "n"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", RowBox[List[TagBox[FractionBox[RowBox[List["k", "+", "n", "+", SubscriptBox["b", "3"], "-", "1"]], "n"], HypergeometricPFQ, Rule[Editable, True]], ";", TagBox[RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", SuperscriptBox["z", "n"]]], HypergeometricPFQ, Rule[Editable, True]]]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, True]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </list> <list> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 3 </cn> </apply> </list> <ci> z </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <cn type='integer'> 3 </cn> </uplimit> <apply> <ci> Pochhammer </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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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,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] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
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