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http://functions.wolfram.com/05.06.06.0037.01
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JacobiP[n, a, b, z] == ((Gamma[1 + a + n] Pochhammer[1 + a + b + n, n])/2^n)
z^n Sum[(((-1)^j/((n - j)! j! Pochhammer[-a - b - 2 n, j]))
Hypergeometric2F1Regularized[-j, -b - n, 1 + a - j + n, -1])/z^j,
{j, 0, n}]
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Cell[BoxData[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "n"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "n"]], ",", "n"]], "]"]]]], SuperscriptBox["2", "n"]], SuperscriptBox["z", "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", "n"]]]], ",", "j"]], "]"]]]]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "j"]], ",", RowBox[List[RowBox[List["-", "b"]], "-", "n"]], ",", RowBox[List["1", "+", "a", "-", "j", "+", "n"]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "j"]]]]]]]]]]]]]
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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> <msubsup> <mi> P </mi> <mi> n </mi> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> n </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", "+", "n", "+", "1"]], ")"]], "n"], Pochhammer] </annotation> </semantics> </mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> - </mo> <mi> b </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "n"]]]], ")"]], "j"], Pochhammer] </annotation> </semantics> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mo> ; </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> j </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", "j"]], Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["-", "b"]], "-", "n"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["a", "-", "j", "+", "n", "+", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", "1"]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> j </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> JacobiP </ci> <ci> n </ci> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <ci> j </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["n_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "+", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "n"]], ",", "n"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List[RowBox[List["-", "j"]], ",", RowBox[List[RowBox[List["-", "b"]], "-", "n"]], ",", RowBox[List["1", "+", "a", "-", "j", "+", "n"]], ",", RowBox[List["-", "1"]]]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "j"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "-", "b", "-", RowBox[List["2", " ", "n"]]]], ",", "j"]], "]"]]]]]]]]], SuperscriptBox["2", "n"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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