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
ChebyshevT






Mathematica Notation

Traditional Notation









Polynomials > ChebyshevT[n,z] > Transformations > Addition formulas





http://functions.wolfram.com/05.04.16.0004.02









  


  










Input Form





ChebyshevT[n, Cos[Subscript[\[Theta], 0]]] == Sum[(n/Pi) 4^k (((n - k)! (1/2 - k))/Gamma[n + k]) Piecewise[{{1, k == Subscript[j, 1] == Subscript[j, 2] == 0}, {0, k + Subscript[j, 1] == 0 || k + Subscript[j, 2] == 0}}, (Gamma[Subscript[j, 1] + k] Gamma[Subscript[j, 2] + k])/ (Gamma[k] Gamma[k])] (Gamma[n - Subscript[j, 1]]/ (Subscript[j, 1]! Gamma[n - k - Subscript[j, 1] + 1])) (Gamma[n - Subscript[j, 2]]/(Subscript[j, 2]! Gamma[n - k - Subscript[j, 2] + 1])) ((Gamma[Subscript[j, 3] - 1/2] Gamma[k - Subscript[j, 3] - 1/2])/ (Subscript[j, 3]! Gamma[k - Subscript[j, 3] + 1])) (1 - (1/2) KroneckerDelta[2 Subscript[j, 1], n - k]) (1 - (1/2) KroneckerDelta[2 Subscript[j, 2], n - k]) (1 - (1/2) KroneckerDelta[2 Subscript[j, 3], k]) Sin[\[Theta]]^k Sin[\[CurlyTheta]]^k ChebyshevT[n - k - 2 Subscript[j, 1], Cos[\[Theta]]] ChebyshevT[n - k - 2 Subscript[j, 2], Cos[\[CurlyTheta]]] ChebyshevT[k - 2 Subscript[j, 3], Cos[\[Phi]]], {k, 0, n}, {Subscript[j, 1], 0, Floor[(n - k)/2]}, {Subscript[j, 2], 0, Floor[(n - k)/2]}, {Subscript[j, 3], 0, Floor[k/2]}] /; Element[n, Integers] && n >= 0 && Cos[Subscript[\[Theta], 0]] == Cos[\[Theta]] Cos[\[CurlyTheta]] + Sin[\[Theta]] Sin[\[CurlyTheta]] Cos[\[Phi]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", RowBox[List["Cos", "[", SubscriptBox["\[Theta]", "0"], "]"]]]], "]"]], "\[Equal]", "\[IndentingNewLine]", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", "0"]], RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["n", "-", "k"]], "2"], "\[RightFloor]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], RowBox[List["\[LeftFloor]", FractionBox[RowBox[List["n", "-", "k"]], "2"], "\[RightFloor]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "3"], "=", "0"]], RowBox[List["\[LeftFloor]", FractionBox["k", "2"], "\[RightFloor]"]]], RowBox[List[FractionBox["n", "\[Pi]"], SuperscriptBox["4", "k"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], RowBox[List["(", RowBox[List[RowBox[List["1", "/", "2"]], "-", "k"]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["n", "+", "k"]], "]"]]], "\[IndentingNewLine]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["1", ",", RowBox[List["k", "\[Equal]", SubscriptBox["j", "1"], "\[Equal]", SubscriptBox["j", "2"], "\[Equal]", "0"]]]], "}"]], ",", " ", RowBox[List["{", " ", RowBox[List["0", ",", RowBox[List[RowBox[List[RowBox[List["k", "+", SubscriptBox["j", "1"]]], "\[Equal]", "0"]], "||", " ", RowBox[List[RowBox[List["k", "+", SubscriptBox["j", "2"]]], "\[Equal]", "0"]]]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["j", "1"], "+", "k"]], "]"]], RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["j", "2"], "+", "k"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "k", "]"]], RowBox[List["Gamma", "[", "k", "]"]]]]]]], "]"]], FractionBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", SubscriptBox["j", "1"]]], "]"]], RowBox[List[RowBox[List[SubscriptBox["j", "1"], "!"]], RowBox[List["Gamma", "[", RowBox[List["n", "-", "k", "-", SubscriptBox["j", "1"], "+", "1"]], "]"]]]]], "\[IndentingNewLine]", FractionBox[RowBox[List["Gamma", "[", RowBox[List["n", "-", SubscriptBox["j", "2"]]], "]"]], RowBox[List[RowBox[List[SubscriptBox["j", "2"], "!"]], RowBox[List["Gamma", "[", RowBox[List["n", "-", "k", "-", SubscriptBox["j", "2"], "+", "1"]], "]"]]]]], FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["j", "3"], "-", RowBox[List["1", "/", "2"]]]], "]"]], RowBox[List["Gamma", "[", RowBox[List["k", "-", SubscriptBox["j", "3"], "-", RowBox[List["1", "/", "2"]]]], "]"]]]], RowBox[List[RowBox[List[SubscriptBox["j", "3"], "!"]], RowBox[List["Gamma", "[", RowBox[List["k", "-", SubscriptBox["j", "3"], "+", "1"]], "]"]]]]], "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["2", SubscriptBox["j", "1"]]], ",", RowBox[List["n", "-", "k"]]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["2", SubscriptBox["j", "2"]]], ",", RowBox[List["n", "-", "k"]]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["2", SubscriptBox["j", "3"]]], ",", "k"]], "]"]]]]]], ")"]], "\[IndentingNewLine]", SuperscriptBox[RowBox[List["Sin", "[", "\[Theta]", "]"]], "k"], SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "k"], "\[IndentingNewLine]", RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["n", "-", "k", "-", RowBox[List["2", SubscriptBox["j", "1"]]]]], ",", RowBox[List["Cos", "[", "\[Theta]", "]"]]]], "]"]], RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["n", "-", "k", "-", RowBox[List["2", SubscriptBox["j", "2"]]]]], ",", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "]"]], RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["k", "-", RowBox[List["2", SubscriptBox["j", "3"]]]]], ",", RowBox[List["Cos", "[", "\[Phi]", "]"]]]], "]"]]]]]]]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List["Cos", "[", SubscriptBox["\[Theta]", "0"], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "\[Theta]", "]"]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", "\[Theta]", "]"]], " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["Cos", "[", "\[Phi]", "]"]]]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='StandardForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#952; </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> k </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mfrac> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msup> <mn> 4 </mn> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mrow> <mi> k </mi> <mo> &#63449; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> &#63449; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mrow> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox[&quot;True&quot;, &quot;PiecewiseDefault&quot;, Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </mrow> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> <mo> , </mo> <mi> k </mi> </mrow> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <mrow> <mrow> <msup> <mi> sin </mi> <mi> k </mi> </msup> <mo> ( </mo> <mi> &#952; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mi> k </mi> </msup> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> T </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 1 </mn> </msub> </mrow> </mrow> </msub> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#952; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> T </mi> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </msub> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> T </mi> <mrow> <mi> k </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> j </mi> <mn> 3 </mn> </msub> </mrow> </mrow> </msub> <mo> ( </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#981; </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> &#952; </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#952; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#981; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#952; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#977; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <times /> <apply> <eq /> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> &#952; </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <pi /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <piecewise> <piece> <cn type='integer'> 1 </cn> <apply> <eq /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 0 </cn> </apply> </piece> <piece> <cn type='integer'> 0 </cn> <apply> <or /> <apply> <eq /> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <otherwise> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> k </ci> </apply> <apply> <ci> Gamma </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </otherwise> </piecewise> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Condition </ci> <apply> <times /> <apply> <power /> <apply> <sin /> <ci> &#952; </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <sin /> <ci> &#977; </ci> </apply> <ci> k </ci> </apply> <apply> <ci> ChebyshevT </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <cos /> <ci> &#952; </ci> </apply> </apply> <apply> <ci> ChebyshevT </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <cos /> <ci> &#977; </ci> </apply> </apply> <apply> <ci> ChebyshevT </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> j </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <cos /> <ci> &#981; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> <apply> <eq /> <apply> <cos /> <apply> <ci> Subscript </ci> <ci> &#952; </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <cos /> <ci> &#952; </ci> </apply> <apply> <cos /> <ci> &#977; </ci> </apply> </apply> <apply> <times /> <apply> <cos /> <ci> &#981; </ci> </apply> <apply> <sin /> <ci> &#952; </ci> </apply> <apply> <sin /> <ci> &#977; </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ChebyshevT", "[", RowBox[List["n_", ",", RowBox[List["Cos", "[", SubscriptBox["\[Theta]_", "0"], "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "1"], "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "k"]], "2"], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "2"], "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "k"]], "2"], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["j", "3"], "=", "0"]], RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]]], FractionBox[RowBox[List["n", " ", SuperscriptBox["4", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[FractionBox["1", "2"], "-", "k"]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Piecewise]", GridBox[List[List["1", RowBox[List["k", "\[Equal]", SubscriptBox["j", "1"], "\[Equal]", SubscriptBox["j", "2"], "\[Equal]", "0"]]], List["0", RowBox[List[RowBox[List[RowBox[List["k", "+", SubscriptBox["j", "1"]]], "\[Equal]", "0"]], "||", RowBox[List[RowBox[List["k", "+", SubscriptBox["j", "2"]]], "\[Equal]", "0"]]]]], List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["j", "1"], "+", "k"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["j", "2"], "+", "k"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", "k", "]"]], " ", RowBox[List["Gamma", "[", "k", "]"]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "-", SubscriptBox["j", "1"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "-", SubscriptBox["j", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[SubscriptBox["j", "3"], "-", FractionBox["1", "2"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", SubscriptBox["j", "3"], "-", FractionBox["1", "2"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "1"]]], ",", RowBox[List["n", "-", "k"]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "2"]]], ",", RowBox[List["n", "-", "k"]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["2", " ", SubscriptBox["j", "3"]]], ",", "k"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "\[Theta]", "]"]], "k"], " ", SuperscriptBox[RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], "k"], " ", RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["n", "-", "k", "-", RowBox[List["2", " ", SubscriptBox["j", "1"]]]]], ",", RowBox[List["Cos", "[", "\[Theta]", "]"]]]], "]"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["n", "-", "k", "-", RowBox[List["2", " ", SubscriptBox["j", "2"]]]]], ",", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "]"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List[RowBox[List["k", "-", RowBox[List["2", " ", SubscriptBox["j", "3"]]]]], ",", RowBox[List["Cos", "[", "\[Phi]", "]"]]]], "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["n", "+", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["j", "1"], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "-", "k", "-", SubscriptBox["j", "1"], "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["j", "2"], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["n", "-", "k", "-", SubscriptBox["j", "2"], "+", "1"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["j", "3"], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", SubscriptBox["j", "3"], "+", "1"]], "]"]]]], ")"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List["Cos", "[", SubscriptBox["\[Theta]", "0"], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "\[Theta]", "]"]], " ", RowBox[List["Cos", "[", "\[CurlyTheta]", "]"]]]], "+", RowBox[List[RowBox[List["Sin", "[", "\[Theta]", "]"]], " ", RowBox[List["Sin", "[", "\[CurlyTheta]", "]"]], " ", RowBox[List["Cos", "[", "\[Phi]", "]"]]]]]]]]]]]]]]]]










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





2002-12-18





© 1998-2014 Wolfram Research, Inc.