Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
JacobiP






Mathematica Notation

Traditional Notation









Polynomials > JacobiP[n,a,b,z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself > Expansions in 1/z





http://functions.wolfram.com/05.06.06.0036.01









  


  










Input Form





JacobiP[n, a, b, z] \[Proportional] (Pochhammer[1 + a + b + n, n]/(2^n n!)) z^n (1 - ((-a + b) n)/((a + b + 2 n) z) + ((a^2 + (-1 + b) b - a (1 + 2 b) - 2 n) (-1 + n) n)/ (2 (-1 + a + b + 2 n) (a + b + 2 n) z^2) + O[1/z^3])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "n"]], ",", "n"]], "]"]]]], RowBox[List["n", "!"]]], SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "n"]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ")"]], " ", "b"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], "-", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", "n"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "3"]], "]"]]]], ")"]], " "]]]]]]










MathML Form







<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> &#8733; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> &#8290; </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[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;, &quot;+&quot;, &quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> JacobiP </ci> <ci> n </ci> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </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> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> <ci> b </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["JacobiP", "[", RowBox[List["n_", ",", "a_", ",", "b_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "a", "+", "b", "+", "n"]], ",", "n"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "b"]], ")"]], " ", "n"]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "b"]], ")"]], " ", "b"]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "b"]]]], ")"]]]], "-", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], " ", "n"]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "3"]], "]"]]]], ")"]]]], RowBox[List["n", "!"]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02





© 1998-2014 Wolfram Research, Inc.