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] > Identities > Recurrence identities > Consecutive neighbors > With respect to n





http://functions.wolfram.com/05.06.17.0001.01









  


  










Input Form





JacobiP[n, a, b, z] == (((3 + a + b + 2 n) (a^2 - b^2 + z (2 + a + b + 2 n) (4 + a + b + 2 n)))/ (2 (1 + a + n) (1 + b + n) (4 + a + b + 2 n))) JacobiP[n + 1, a, b, z] - (((2 + n) (2 + a + b + n) (2 + a + b + 2 n))/((1 + a + n) (1 + b + n) (4 + a + b + 2 n))) JacobiP[n + 2, a, b, z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["JacobiP", "[", RowBox[List["n", ",", "a", ",", "b", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["3", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]]]], ")"]]]], RowBox[List["JacobiP", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "a", ",", "b", ",", "z"]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], RowBox[List["JacobiP", "[", RowBox[List[RowBox[List["n", "+", "2"]], ",", "a", ",", "b", ",", "z"]], "]"]]]]]]]]]]










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> &#10869; </mo> <mrow> <mrow> <mfrac> <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> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mi> z </mi> <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> 2 </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> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> n </mi> <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> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> P </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <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> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </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> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mi> n </mi> <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> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msubsup> <mi> P </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <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> </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> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <ci> n </ci> <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> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> JacobiP </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> b </ci> <ci> n </ci> <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> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> JacobiP </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> <ci> b </ci> <ci> z </ci> </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]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["3", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]]], ")"]]]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", "a", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]]]], ")"]], " ", RowBox[List["JacobiP", "[", RowBox[List[RowBox[List["n", "+", "2"]], ",", "a", ",", "b", ",", "z"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "a", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", "b", "+", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "a", "+", "b", "+", RowBox[List["2", " ", "n"]]]], ")"]]]]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.