Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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
ChebyshevT




Mathematica Notation

Traditional Notation









Polynomials > ChebyshevT[n,z] > Summation > Infinite summation





http://functions.wolfram.com/05.04.23.0006.01









  


  








Input Form





Sum[((Pochhammer[\[Gamma], n] Pochhammer[-\[Gamma], n])/ (Pochhammer[1/2, n] n!)) ChebyshevT[n, z] w^n, {n, 0, Infinity}] == Cos[2 \[Gamma] ArcSin[Sqrt[1 - w - Sqrt[1 + w^2 - 2 w z]]/Sqrt[2]]] Cos[2 \[Gamma] ArcSin[Sqrt[1 + w - Sqrt[1 + w^2 - 2 w z]]/Sqrt[2]]] /; -1 < z < 1 && Abs[w] < 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Gamma]", ",", "n"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Gamma]"]], ",", "n"]], "]"]], " "]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n"]], "]"]], RowBox[List["n", "!"]]]]], RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", "z"]], "]"]], " ", SuperscriptBox["w", "n"]]]]], "\[Equal]", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Gamma]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "w", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["w", "2"], "-", RowBox[List["2", " ", "w", " ", "z"]]]]]]]], SqrtBox["2"]], "]"]]]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Gamma]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "+", "w", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["w", "2"], "-", RowBox[List["2", " ", "w", " ", "z"]]]]]]]], SqrtBox["2"]], "]"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "z", "<", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> &#947; </mi> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;\[Gamma]&quot;, &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#947; </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;\[Gamma]&quot;]], &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mi> n </mi> </msup> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#947; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> w </mi> </mrow> <mo> - </mo> <msqrt> <mrow> <msup> <mi> w </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#947; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mi> w </mi> <mo> - </mo> <msqrt> <mrow> <msup> <mi> w </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &lt; </mo> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> w </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <ci> &#947; </ci> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#947; </ci> </apply> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 2 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <power /> <ci> w </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#947; </ci> <apply> <arcsin /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> w </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <ci> w </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#947; </ci> <apply> <arcsin /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> w </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <ci> w </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <cn type='integer'> -1 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <lt /> <apply> <abs /> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["\[Gamma]_", ",", "n_"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Gamma]_"]], ",", "n_"]], "]"]]]], ")"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List["n_", ",", "z_"]], "]"]], " ", SuperscriptBox["w_", "n_"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "n_"]], "]"]], " ", RowBox[List["n_", "!"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Gamma]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "-", "w", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["w", "2"], "-", RowBox[List["2", " ", "w", " ", "z"]]]]]]]], SqrtBox["2"]], "]"]]]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "\[Gamma]", " ", RowBox[List["ArcSin", "[", FractionBox[SqrtBox[RowBox[List["1", "+", "w", "-", SqrtBox[RowBox[List["1", "+", SuperscriptBox["w", "2"], "-", RowBox[List["2", " ", "w", " ", "z"]]]]]]]], SqrtBox["2"]], "]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "<", "z", "<", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", "w", "]"]], "<", "1"]]]]]]]]]]


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





2001-10-29