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http://functions.wolfram.com/07.23.16.0007.01
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Hypergeometric2F1[a, b, c, g z] Hypergeometric2F1[\[Alpha], \[Beta],
\[Gamma], h z] == Sum[((Pochhammer[a, m] Pochhammer[b, m] g^m)/
(Pochhammer[c, m] m!)) ((Pochhammer[\[Alpha], k - m]
Pochhammer[\[Beta], k - m] h^(k - m))/(Pochhammer[\[Gamma], k - m]
(k - m)!)) z^k, {k, 0, Infinity}, {m, 0, k}]
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Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a", ",", "b", ",", "c", ",", RowBox[List["g", " ", "z"]]]], "]"]], RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", "\[Beta]", ",", "\[Gamma]", ",", RowBox[List["h", " ", "z"]]]], "]"]]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "m"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List["b", ",", "m"]], "]"]], " ", SuperscriptBox["g", "m"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["c", ",", "m"]], "]"]], " ", RowBox[List["m", "!"]]]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Alpha]", ",", RowBox[List["k", "-", "m"]]]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List["\[Beta]", ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", SuperscriptBox["h", RowBox[List["k", "-", "m"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Gamma]", ",", RowBox[List["k", "-", "m"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]]]]], SuperscriptBox["z", "k"]]]]]]]]]]]]]
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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> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ; </mo> <mi> c </mi> <mo> ; </mo> <mrow> <mi> g </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["a", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["b", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["c", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["g", " ", "z"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> α </mi> <mo> , </mo> <mi> β </mi> </mrow> <mo> ; </mo> <mi> γ </mi> <mo> ; </mo> <mrow> <mi> h </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["\[Alpha]", Hypergeometric2F1, Rule[Editable, True]], ",", TagBox["\[Beta]", Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox["\[Gamma]", Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], ";", TagBox[RowBox[List["h", " ", "z"]], Hypergeometric2F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "a", ")"]], "m"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "b", ")"]], "m"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Alpha]", ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> β </mi> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Beta]", ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> g </mi> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mi> h </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> c </mi> <mo> ) </mo> </mrow> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "c", ")"]], "m"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> γ </mi> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Gamma]", ")"]], RowBox[List["k", "-", "m"]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <ci> Hypergeometric2F1 </ci> <ci> a </ci> <ci> b </ci> <ci> c </ci> <apply> <times /> <ci> g </ci> <ci> z </ci> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <ci> α </ci> <ci> β </ci> <ci> γ </ci> <apply> <times /> <ci> h </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <ci> a </ci> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> b </ci> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> α </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <ci> β </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <ci> g </ci> <ci> m </ci> </apply> <apply> <power /> <ci> h </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <ci> c </ci> <ci> m </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> γ </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["a_", ",", "b_", ",", "c_", ",", RowBox[List["g_", " ", "z_"]]]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]_", ",", "\[Beta]_", ",", "\[Gamma]_", ",", RowBox[List["h_", " ", "z_"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "m"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["b", ",", "m"]], "]"]], " ", SuperscriptBox["g", "m"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Alpha]", ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Beta]", ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", SuperscriptBox["h", RowBox[List["k", "-", "m"]]]]], ")"]], " ", SuperscriptBox["z", "k"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["c", ",", "m"]], "]"]], " ", RowBox[List["m", "!"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Gamma]", ",", RowBox[List["k", "-", "m"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "m"]], ")"]], "!"]]]], ")"]]]]]]]]]]]]] |
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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,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] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
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