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variants of this functions
ChebyshevU






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ChebyshevU[nu,z] > Summation > Infinite summation





http://functions.wolfram.com/07.05.23.0003.01









  


  










Input Form





Sum[(1/(Pochhammer[3/2, n] (n + 1)!)) ChebyshevU[n, z] w^n, {n, 0, Infinity}] == (Sinh[Sqrt[2] Sqrt[w (-1 + z)]] Sinh[Sqrt[2] Sqrt[w (1 + z)]])/(2 Sqrt[w (-1 + z)] Sqrt[w (1 + z)]) /; -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["1", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "n"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]], RowBox[List["ChebyshevU", "[", RowBox[List["n", ",", "z"]], "]"]], SuperscriptBox["w", "n"]]]]], "\[Equal]", " ", FractionBox[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]]]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]]], " ", SqrtBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]]]]]]]]]], "/;", 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> <mfrac> <mrow> <mrow> <msub> <mi> U </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> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[&quot;3&quot;, &quot;2&quot;], &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> </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> <ci> ChebyshevU </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <power /> <ci> w </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <cn type='rational'> 3 <sep /> 2 </cn> <ci> n </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <sinh /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <sinh /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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["ChebyshevU", "[", RowBox[List["n_", ",", "z_"]], "]"]], " ", SuperscriptBox["w_", "n_"]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "n_"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n_", "+", "1"]], ")"]], "!"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sinh", "[", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]]]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]]], " ", SqrtBox[RowBox[List["w", " ", RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]]]]]]]], "/;", 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





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