Jacobi: Special cases

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Jacobi: Special cases

Koornwinder Addendum: Jacobi

Orthogonality relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0 (\Jacobi{\alpha}{\beta}{n}@{1})^2}= \frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1} \frac{\pochhammer{\beta+1}{n} n!}{\pochhammer{\alpha+1}{n} \pochhammer{\alpha+\beta+2}{n}} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0}= \frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1} \frac{\pochhammer{\alpha+1}{n}\pochhammer{\beta+1}{n}}{\pochhammer{\alpha+\beta+2}{n} n!} h_0 = \frac{n+\alpha+\beta+1}{2n+\alpha+\beta+1} \frac{\pochhammer{\alpha+1}{n}\pochhammer{\beta+1}{n}}{\pochhammer{\alpha+\beta+2}{n} n!} h_0}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0} =\frac{2^{\alpha+\beta+1}\EulerGamma@{\alpha+1}\EulerGamma@{\beta+1}}{\EulerGamma@{\alpha+\beta+2}} =\frac{2^{\alpha+\beta+1}\EulerGamma@{\alpha+1}\EulerGamma@{-\alpha-\beta-1}}{\EulerGamma@{-\beta}}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_1^\infty \Jacobi{\alpha}{\beta}{m}@{x} \Jacobi{\alpha}{\beta}{n}@{x} (x-1)^\alpha (x+1)^\beta dx=h_n \Kronecker{m}{n} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle -1-\beta>\alpha>-1, m,n<-\frac12(\alpha+\beta+1) }}

Symmetry

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Jacobi{\alpha}{\beta}{n}@{-x}=(-1)^n \Jacobi{\beta}{\alpha}{n}@{x} }}

Jacobi: Special cases: Special values

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Jacobi{\alpha}{\beta}{n}@{1}=\frac{\pochhammer{\alpha+1}{n}}{n!} \Jacobi{\alpha}{\beta}{n}@{-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Jacobi{\alpha}{\beta}{n}@{1} =\frac{(-1)^n\pochhammer{\beta+1}{n}}{n!} \frac{\Jacobi{\alpha}{\beta}{n}@{-1}}{\Jacobi{\alpha}{\beta}{n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Jacobi{\alpha}{\beta}{n}@{1} =\frac{(-1)^n\pochhammer{\beta+1}{n}}{\pochhammer{\alpha+1}{n}} }}

Bilateral generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{n=0}^\infty\frac{\pochhammer{\alpha+\beta+1}{n} n!}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+1}{n}} r^n \Jacobi{\alpha}{\beta}{n}@{x} \Jacobi{\alpha}{\beta}{n}@{y} =\frac1{(1+r)^{\alpha+\beta+1}} \AppellFiv@{\frac12(\alpha+\beta+1)}{\frac12(\alpha+\beta+2)}{\alpha+1}{\beta+1}{ \frac{r(1-x)(1-y)}{(1+r)^2}}{\frac{r(1+x)(1+y)}{(1+r)^2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{n=0}^\infty\frac{2n+\alpha+\beta+1}{n+\alpha+\beta+1} \frac{\pochhammer{\alpha+\beta+2}{n} n!}{\pochhammer{\alpha+1}{n}\pochhammer{\beta+1}{n}} r^n \Jacobi{\alpha}{\beta}{n}@{x} \Jacobi{\alpha}{\beta}{n}@{y} =\frac{1-r}{(1+r)^{\alpha+\beta+2}} \AppellFiv@{\frac12(\alpha+\beta+2)}{\frac12(\alpha+\beta+3)}{\alpha+1}{\beta+1}{ \frac{r(1-x)(1-y)}{(1+r)^2}}{\frac{r(1+x)(1+y)}{(1+r)^2}} }}

Quadratic transformations

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Ultra{\alpha+\frac12}{2n}@{x}}{\Ultra{\alpha+\frac12}{2n}@{1}} =\frac{\Jacobi{\alpha}{\alpha}{2n}@{x}}{\Jacobi{\alpha}{\alpha}{2n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Ultra{\alpha+\frac12}{2n}@{x}}{\Ultra{\alpha+\frac12}{2n}@{1}} =\frac{\Jacobi{\alpha}{-\frac12}{n}@{2x^2-1}}{\Jacobi{\alpha}{-\frac12}{n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Ultra{\alpha+\frac12}{2n+1}@{x}}{\Ultra{\alpha+\frac12}{2n+1}@{1}} =\frac{\Jacobi{\alpha}{\alpha}{2n+1}@{x}}{\Jacobi{\alpha}{\alpha}{2n+1}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Ultra{\alpha+\frac12}{2n+1}@{x}}{\Ultra{\alpha+\frac12}{2n+1}@{1}} =\frac{x \Jacobi{\alpha}{\frac12}{n}@{2x^2-1}}{\Jacobi{\alpha}{\frac12}{n}@{1}} }}

Differentiation formulas

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac d{dx}\left((1-x)^\alpha \Jacobi{\alpha}{\beta}{n}@{x}\right)= -(n+\alpha) (1-x)^{\alpha-1} \Jacobi{\alpha-1}{\beta+1}{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left((1-x)\frac d{dx}-\alpha\right)\Jacobi{\alpha}{\beta}{n}@{x}= -(n+\alpha) \Jacobi{\alpha-1}{\beta+1}{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac d{dx}\left((1+x)^\beta \Jacobi{\alpha}{\beta}{n}@{x}\right)= (n+\beta) (1+x)^{\beta-1} \Jacobi{\alpha+1}{\beta-1}{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left((1+x)\frac d{dx}+\beta\right)\Jacobi{\alpha}{\beta}{n}@{x}= (n+\beta) \Jacobi{\alpha+1}{\beta-1}{n}@{x} }}

Generalized Gegenbauer polynomials

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenGegenbauer{\alpha}{\beta}{2m}@{x}:=\mathrm const \Jacobi{\alpha}{\beta}{m}@{2x^2-1},\qquad \GenGegenbauer{\alpha}{\beta}{2m+1}@{x}: }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenGegenbauer{\alpha}{\beta}{2m}@{x} =\mathrm const x \Jacobi{\alpha}{\beta+1}{m}@{2x^2-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-1}^1 \GenGegenbauer{\alpha}{\beta}{m}@{x} \GenGegenbauer{\alpha}{\beta}{n}@{x} |x|^{2\beta+1}(1-x^2)^\alpha dx =0\qquad(m\ne n) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (T_\mu f)(x):=f'(x)+\mu \frac{f(x)-f(-x)}x }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenGegenbauer{\alpha}{\beta}{2m}@{x}=\frac{\pochhammer{\alpha+\beta+1}{m}}{\pochhammer{\beta+1}{m}} \Jacobi{\alpha}{\beta}{m}@{2x^2-1} \GenGegenbauer{\alpha}{\beta}{2m+1}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenGegenbauer{\alpha}{\beta}{2m}@{x} =\frac{\pochhammer{\alpha+\beta+1}{m+1}}{\pochhammer{\beta+1}{m+1}} x \Jacobi{\alpha}{\beta+1}{m}@{2x^2-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle T_{\beta+\frac12}\GenGegenbauer{\alpha}{\beta}{n}=2(\alpha+\beta+1) \GenGegenbauer{\alpha+1}{\beta}{n-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle T_{\beta+\frac12}^2\GenGegenbauer{\alpha}{\beta}{n}=4(\alpha+\beta+1)(\alpha+\beta+2) \GenGegenbauer{\alpha+2}{\beta}{n-2} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left(\frac{d^2}{dx^2}+\frac{2\beta+1}x \frac d{dx}\right)\Jacobi{\alpha}{\beta}{n}@{2x^2-1} =4(n+\alpha+\beta+1)(n+\beta) \Jacobi{\alpha+2}{\beta}{n-1}@{2x^2-1} }}

Gegenbauer / Ultraspherical

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{x}=\frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+\frac{1}{2}}{n}} \Jacobi{\lambda-\frac{1}{2}}{\lambda-\frac{1}{2}}{n}@{x} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda\neq 0}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{x}=\frac{\pochhammer{2\lambda}{n}}{n!}\,\HyperpFq{2}{1}@@{-n,n+2\lambda}{\lambda+\frac{1}{2}}{\frac{1-x}{2}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-1}^1(1-x^2)^{\lambda-\frac{1}{2}}\Ultra{\lambda}{m}@{x}\Ultra{\lambda}{n}@{x}\,dx {}=\frac{\cpi\EulerGamma@{n+2\lambda}2^{1-2\lambda}}{\left\{\EulerGamma@{\lambda}\right\}^2(n+\lambda)n!}\,\Kronecker{m}{n} }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda>-\frac{1}{2}\quad\lambda\neq 0}}


Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2(n+\lambda)x\Ultra{\lambda}{n}@{x}=(n+1)\Ultra{\lambda}{n+1}@{x}+(n+2\lambda-1)\Ultra{\lambda}{n-1}@{x} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicUltra{\lambda}{n}@@{x}{x}=\monicUltra{\lambda}{n+1}@@{x}{x}+\frac{n(n+2\lambda-1)}{4(n+\lambda-1)(n+\lambda)}\monicUltra{\lambda}{n-1}@@{x}{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{x}=\frac{2^n\pochhammer{\lambda}{n}}{n!}\monicUltra{\lambda}{n}@@{x}{x} }}

Differential equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)y''(x)-(2\lambda+1)xy'(x)+n(n+2\lambda)y(x)=0 }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\Ultra{\lambda}{n}@{x}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\Ultra{\lambda}{n}@{x}=2\lambda \Ultra{\lambda+1}{n-1}@{x} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)\frac{d}{dx}\Ultra{\lambda}{n}@{x}+(1-2\lambda)x\Ultra{\lambda}{n}@{x}= -\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)} \Ultra{\lambda-1}{n+1}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\left[(1-x^2)^{\lambda-\textstyle\frac{1}{2}}\Ultra{\lambda}{n}@{x}\right] {}=-\frac{(n+1)(2\lambda+n-1)}{2(\lambda-1)}(1-x^2)^{\lambda-\textstyle\frac{3}{2}}\Ultra{\lambda-1}{n+1}@{x} }}

Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)^{\lambda-\frac{1}{2}}\Ultra{\lambda}{n}@{x}= \frac{\pochhammer{2\lambda}{n}(-1)^n}{\pochhammer{\lambda+\frac{1}{2}}{n}2^nn!}\left(\frac{d}{dx}\right)^n \left[(1-x^2)^{\lambda+n-\frac{1}{2}}\right] }}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-2xt+t^2)^{-\lambda}=\sum_{n=0}^{\infty}\Ultra{\lambda}{n}@{x}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R^{-1}\left(\frac{1+R-xt}{2}\right)^{\frac{1}{2}-\lambda}=\sum_{n=0}^{\infty} \frac{\pochhammer{\lambda+\frac{1}{2}}{n}}{\pochhammer{2\lambda}{n}}\Ultra{\lambda}{n}@{x}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R=\sqrt{1-2xt+t^2}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x+1)t}{2}} =\sum_{n=0}^{\infty}\frac{\Ultra{\lambda}{n}@{x}} {\pochhammer{2\lambda}{n}\pochhammer{\lambda+\frac{1}{2}}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{xt}\,\HyperpFq{0}{1}@@{-}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= \sum_{n=0}^{\infty}\frac{\Ultra{\lambda}{n}@{x}}{\pochhammer{2\lambda}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,2\lambda-\gamma}{\lambda+\frac{1}{2}}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{2\lambda-\gamma}{n}} {\pochhammer{2\lambda}{n}\pochhammer{\lambda+\frac{1}{2}}{n}}\Ultra{\lambda}{n}@{x}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R=\sqrt{1-2xt+t^2}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\lambda+\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{\pochhammer{2\lambda}{n}}\Ultra{\lambda}{n}@{x}t^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Limit relation

Gegenbauer / Ultraspherical polynomial to Hermite polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\alpha\rightarrow\infty} \alpha^{-\frac{1}{2}n}\Ultra{\alpha+\frac{1}{2}}{n}@{\alpha^{-\frac{1}{2}}x}=\frac{\Hermite{n}@{x}}{n!} }}

Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{2n}@{x}=\frac{\pochhammer{\lambda}{n}}{\pochhammer{\frac{1}{2}}{n}} \Jacobi{\lambda-\frac{1}{2}}{-\frac{1}{2}}{n}@{2x^2-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{2n+1}@{x}=\frac{\pochhammer{\lambda}{n+1}}{\pochhammer{\frac{1}{2}}{n+1}} x\Jacobi{\lambda-\frac{1}{2}}{\frac{1}{2}}{n}@{2x^2-1} }}

Koornwinder Addendum: Gegenbauer

Orthogonality relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0}= \frac\lambda{\lambda+n} \frac{\pochhammer{2\lambda}{n}}{n!} h_0 }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{h_n}{h_0} =\frac{\cpi^\frac12 \EulerGamma@{\lambda+\frac12}}{\EulerGamma@{\lambda+1}} \frac{h_n}{h_0 (\Ultra{\lambda}{n}@{1})^2} = \frac\lambda{\lambda+n} \frac{n!}{\pochhammer{2\lambda}{n}}}}


Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{x}=\sum_{\ell=0}^{\lfloor n/2\rfloor}\frac{(-1)^{\ell}\pochhammer{\lambda}{n-\ell}} {\ell!\;(n-2\ell)!} (2x)^{n-2\ell} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{x} =(2x)^{n} \frac{\pochhammer{\lambda}{n}}{n!} \HyperpFq{2}{1}@@{-\frac12 n,-\frac12 n+\frac12}{1-\lambda-n}{\frac1{x^2}} }}

Jacobi: Special cases: Special value

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{1}=\frac{\pochhammer{2\lambda}{n}}{n!} }}

Expression in terms of Jacobi

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Ultra{\lambda}{n}@{x}}{\Ultra{\lambda}{n}@{1}}= \frac{\Jacobi{\lambda-\frac12}{\lambda-\frac12}{n}@{x}}{\Jacobi{\lambda-\frac12}{\lambda-\frac12}{n}@{1}} ,\qquad \Ultra{\lambda}{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\Ultra{\lambda}{n}@{x}}{\Ultra{\lambda}{n}@{1}} =\frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+\frac12}{n}} \Jacobi{\lambda-\frac12}{\lambda-\frac12}{n}@{x} }}

Re: (9.8.21)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x^2 \Ultra{\lambda}{n}@{x}= \frac{(n+1)(n+2)}{4(n+\lambda)(n+\lambda+1)} \Ultra{\lambda}{n+2}@{x}+ \frac{n^2+2n\lambda+\lambda-1}{2(n+\lambda-1)(n+\lambda+1)} \Ultra{\lambda}{n}@{x} +\frac{(n+2\lambda-1)(n+2\lambda-2)}{4(n+\lambda)(n+\lambda-1)} \Ultra{\lambda}{n-2}@{x} }}

Bilateral generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{n=0}^\infty\frac{n!}{\pochhammer{2\lambda}{n}} r^n \Ultra{\lambda}{n}@{x} \Ultra{\lambda}{n}@{y} =\frac1{(1-2rxy+r^2)^\lambda} \HyperpFq{2}{1}@@{\frac12\lambda,\frac12(\lambda+1)}{\lambda+\frac12}{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}} (r\in(-1,1),\;x,y\in[-1,1]) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{n=0}^\infty\frac{\lambda+n}\lambda \frac{n!}{\pochhammer{2\lambda}{n}} r^n \Ultra{\lambda}{n}@{x} \Ultra{\lambda}{n}@{y} =\frac{1-r^2}{(1-2rxy+r^2)^{\lambda+1}} \HyperpFq{2}{1}@@{\frac12(\lambda+1),\frac12(\lambda+2)}{\lambda+\frac12}{\frac{4r^2(1-x^2)(1-y^2)}{(1-2rxy+r^2)^2}} (r\in(-1,1),\;x,y\in[-1,1]) }}

Trigonometric expansions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\sum_{k=0}^n\frac{\pochhammer{\lambda}{k}\pochhammer{\lambda}{n-k}}{k! (n-k)!} \expe^{\iunit(n-2k)\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\expe^{\iunit n\theta}\frac{\pochhammer{\lambda}{n}}{n!} \HyperpFq{2}{1}@@{-n,\lambda}{1-\lambda-n}{\expe^{-2\iunit\theta}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{\pochhammer{\lambda}{n}}{2^\lambda n!} \expe^{-\frac12 \iunit\lambda\cpi}\expe^{\iunit(n+\lambda)\theta} (\sin@@{\theta})^{-\lambda} \HyperpFq{2}{1}@@{\lambda,1-\lambda}{1-\lambda-n}{\frac{\iunit \expe^{-\iunit\theta}}{2\sin@@{\theta}}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{\pochhammer{\lambda}{n}}{n!} \sum_{k=0}^\infty\frac{\pochhammer{\lambda}{k}\pochhammer{1-\lambda}{k}}{\pochhammer{1-\lambda-n}{k} k!} \frac{\cos@{(n-k+\lambda)\theta+\frac12(k-\lambda)\cpi}}{(2\sin@@{\theta})^{k+\lambda}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}}=\frac{2\EulerGamma@{\lambda+\frac12}}{\cpi^\frac12\EulerGamma@{\lambda+1}} \frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+1}{n}} (\sin@@{\theta})^{1-2\lambda} \sum_{k=0}^\infty\frac{\pochhammer{1-\lambda}{k}\pochhammer{n+1}{k}}{\pochhammer{n+\lambda+1}{k} k!} \sin@{(2k+n+1)\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{2\EulerGamma@{\lambda+\frac12}}{\cpi^\frac12\EulerGamma@{\lambda+1}} \frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+1}{n}} (\sin@@{\theta})^{1-2\lambda} \imagpart{\expe^{\iunit(n+1)\theta} \HyperpFq{2}{1}@@{1-\lambda,n+1}{n+\lambda+1}{\expe^{2\iunit\theta}}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{2^\lambda\EulerGamma@{\lambda+\frac12}}{\cpi^\frac12\EulerGamma@{\lambda+1}} \frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+1}{n}} (\sin@@{\theta})^{-\lambda} \realpart{\expe^{-\frac12 \iunit\lambda\cpi}\expe^{\iunit(n+\lambda)\theta} \HyperpFq{2}{1}@@{\lambda,1-\lambda}{1+\lambda+n}{\frac{\expe^{\iunit\theta}}{2\iunit\sin@@{\theta}}}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Ultra{\lambda}{n}@{\cos@@{\theta}} =\frac{2^{2\lambda}\EulerGamma@{\lambda+\frac12}}{\cpi^\frac12\EulerGamma@{\lambda+1}} \frac{\pochhammer{2\lambda}{n}}{\pochhammer{\lambda+1}{n}} \sum_{k=0}^\infty\frac{\pochhammer{\lambda}{k}\pochhammer{1-\lambda}{k}}{\pochhammer{1+\lambda+n}{k} k!} \frac{\cos@{(n+k+\lambda)\theta-\frac12(k+\lambda)\cpi}}{(2\sin@@{\theta})^{k+\lambda}} }}

Fourier transform

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\EulerGamma@{\lambda+1}}{\EulerGamma@{\lambda+\frac12} \EulerGamma@{\frac12}} \int_{-1}^1 \frac{\Ultra{\lambda}{n}@{y}}{\Ultra{\lambda}{n}@{1}} (1-y^2)^{\lambda-\frac12} \expe^{\iunit xy} dy =\iunit^n 2^\lambda \EulerGamma@{\lambda+1} x^{-\lambda} \BesselJ{\lambda+n}@{x} }}

Laplace transforms

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac2{n! \EulerGamma@{\lambda}} \int_0^\infty \Hermite{n}@{tx} t^{n+2\lambda-1} \expe^{-t^2} dt=\Ultra{\lambda}{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac2{\EulerGamma@{\lambda+\frac12}} \int_0^1 \frac{\Ultra{\lambda}{n}@{t}}{\Ultra{\lambda}{n}@{1}} (1-t^2)^{\lambda-\frac12} t^{-1} (x/t)^{n+2\lambda+1} \expe^{-x^2/t^2} dt =2^{-n} \Hermite{n}@{x} \expe^{-x^2} (\lambda>-\frac12) }}

Addition formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \normJacobiR{\alpha}{\alpha}{n}@{xy+(1-x^2)^\frac12(1-y^2)^\frac12 t} =\sum_{k=0}^n \frac{(-1)^k\pochhammer{-n}{k} \pochhammer{n+2\alpha+1}{k}}{2^{2k}(\pochhammer{\alpha+1}{k})^2} (1-x^2)^{k/2} \normJacobiR{\alpha+k}{\alpha+k}{n-k}@{x} (1-y^2)^{k/2} \normJacobiR{\alpha+k}{\alpha+k}{n-k}@{y} \omega_k^{(\alpha-\frac12,\alpha-\frac12)} \normJacobiR{\alpha-\frac12}{\alpha-\frac12}{k}@{t} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \normJacobiR{\alpha}{\beta}{n}@{x}:=\Jacobi{\alpha}{\beta}{n}@{x}/\Jacobi{\alpha}{\beta}{n}@{1} \omega_n^{(\alpha,\beta)}:=\frac{\int_{-1}^1 (1-x)^\alpha(1+x)^\beta dx} {\int_{-1}^1 (\normJacobiR{\alpha}{\beta}{n}@{x})^2 (1-x)^\alpha(1+x)^\beta dx} }}

Chebyshev

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyT{n}@{x}=\frac{\Jacobi{-\frac{1}{2}}{-\frac{1}{2}}{n}@{x}}{\Jacobi{-\frac{1}{2}}{-\frac{1}{2}}{n}@{1}} =\HyperpFq{2}{1}@@{-n,n}{\frac{1}{2}}{\frac{1-x}{2}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyU{n}@{x}=(n+1)\frac{\Jacobi{\frac{1}{2}}{\frac{1}{2}}{n}@{x}}{\Jacobi{\frac{1}{2}}{\frac{1}{2}}{n}@{1}} =(n+1)\,\HyperpFq{2}{1}@@{-n,n+2}{\frac{3}{2}}{\frac{1-x}{2}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-1}^1(1-x^2)^{-\frac{1}{2}}\ChebyT{m}@{x}\ChebyT{n}@{x}\,dx= \left\{\begin{array}{ll} \displaystyle\frac{\cpi}{2}\,\Kronecker{m}{n}, & n\neq 0\\[5mm] \cpi\,\Kronecker{m}{n}, & n=0 \end{array}\right. }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-1}^1(1-x^2)^{\frac{1}{2}}\ChebyU{m}@{x}\ChebyU{n}@{x}\,dx=\frac{\cpi}{2}\,\Kronecker{m}{n} }}

Recurrence relations

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2x\ChebyT{n}@{x}=\ChebyT{n+1}@{x}+\ChebyT{n-1}@{x},\quad \ChebyT{0}@{x}=1\quad\textrm{and}\quad \ChebyT{1}@{x}=x }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2x\ChebyU{n}@{x}=\ChebyU{n+1}@{x}+\ChebyU{n-1}@{x},\quad \ChebyU{0}@{x}=1\quad\textrm{and}\quad \ChebyU{1}@{x}=2x }}

Monic recurrence relations

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicChebyT{n}@@{x}=\monicChebyT{n+1}@@{x}+\frac{1}{4}\monicChebyT{n-1}@@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyT{1}@{x}=\monicChebyT{1}@@{x}=x\quad\textrm{and}\quad \ChebyT{n}@{x}=2^n\monicChebyT{n}@@{x},\quad n\neq 1 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicChebyU{n}@@{x}=\monicChebyU{n+1}@@{x}+\frac{1}{4}\monicChebyU{n-1}@@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyU{n}@{x}=2^n\monicChebyU{n}@@{x} }}

Differential equations

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)y''(x)-xy'(x)+n^2y(x)=0 }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\ChebyT{n}@{x}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)y''(x)-3xy'(x)+n(n+2)y(x)=0 }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\ChebyU{n}@{x}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\ChebyT{n}@{x}=n\ChebyU{n-1}@{x} }}

Backward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)\frac{d}{dx}\ChebyU{n}@{x}-x\ChebyU{n}@{x}=-(n+1)\ChebyT{n+1}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{d}{dx}\left[\left(1-x^2\right)^{\frac{1}{2}}\ChebyU{n}@{x}\right] =-(n+1)\left(1-x^2\right)^{-\frac{1}{2}}\ChebyT{n+1}@{x} }}

Rodrigues-type formulas

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)^{-\frac{1}{2}}\ChebyT{n}@{x}=\frac{(-1)^n}{\pochhammer{\frac{1}{2}}{n}2^n} \left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n-\frac{1}{2}}\right] }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)^{\frac{1}{2}}\ChebyU{n}@{x}=\frac{(n+1)(-1)^n}{\pochhammer{\frac{3}{2}}{n}2^n} \left(\frac{d}{dx}\right)^n\left[(1-x^2)^{n+\frac{1}{2}}\right] }}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1-xt}{1-2xt+t^2}=\sum_{n=0}^{\infty}\ChebyT{n}@{x}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R^{-1}\sqrt{\frac{1}{2}(1+R-xt)}=\sum_{n=0}^{\infty} \frac{\pochhammer{\frac{1}{2}}{n}}{n!}\ChebyT{n}@{x}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R=\sqrt{1-2xt+t^2}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{0}{1}@@{-}{\frac{1}{2}}{\frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\frac{1}{2}}{\frac{(x+1)t}{2}}= \sum_{n=0}^{\infty}\frac{\ChebyT{n}@{x}}{\pochhammer{\frac{1}{2}}{n}n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{xt}\,\HyperpFq{0}{1}@@{-}{\frac{1}{2}}{\frac{(x^2-1)t^2}{4}}= \sum_{n=0}^{\infty}\frac{\ChebyT{n}@{x}}{n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{2}{1}@@{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,-\gamma}{\frac{1}{2}}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{-\gamma}{n}}{\pochhammer{\frac{1}{2}}{n}n!}\ChebyT{n}@{x}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R=\sqrt{1-2xt+t^2}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{1}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}} =\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{n!}\ChebyT{n}@{x}t^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{1-2xt+t^2}=\sum_{n=0}^{\infty}\ChebyU{n}@{x}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{R\sqrt{\frac{1}{2}(1+R-xt)}}=\sum_{n=0}^{\infty} \frac{\pochhammer{\frac{3}{2}}{n}}{(n+1)!}\ChebyU{n}@{x}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R=\sqrt{1-2xt+t^2}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{0}{1}@@{-}{\frac{3}{2}}{\frac{(x-1)t}{2}}\ \HyperpFq{0}{1}@@{-}{\frac{3}{2}}{\frac{(x+1)t}{2}}= \sum_{n=0}^{\infty}\frac{\ChebyU{n}@{x}}{\pochhammer{\frac{3}{2}}{n}(n+1)!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{xt}\,\HyperpFq{0}{1}@@{-}{\frac{3}{2}}{\frac{(x^2-1)t^2}{4}}= \sum_{n=0}^{\infty}\frac{\ChebyU{n}@{x}}{(n+1)!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{2}{1}@@{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,2-\gamma}{\frac{3}{2}}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{2-\gamma}{n}}{\pochhammer{\frac{3}{2}}{n}(n+1)!}\ChebyU{n}@{x}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R=\sqrt{1-2xt+t^2}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{\frac{3}{2}}{\frac{(x^2-1)t^2}{(1-xt)^2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{(n+1)!}\ChebyU{n}@{x}t^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyT{n}@{x}=\cos@{n\theta} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyU{n}@{x}=\frac{\sin@{n+1}\theta}{\sin@@{\theta}} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyU{n}@{x}=\Ultra{1}{n}@{x} }}

Koornwinder Addendum: Chebyshev

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyT{n}@{x}:=\frac{\Jacobi{-\frac12}{-\frac12}{n}@{x}}{\Jacobi{-\frac12}{-\frac12}{n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyT{n}@{x} =\cos@{n\theta}, x=\cos@@{\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyU{n}@{x}:=(n+1) \frac{\Jacobi{\frac12}{\frac12}{n}@{x}}{\Jacobi{\frac12}{\frac12}{n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyU{n}@{x} =\frac{\sin@{(n+1)\theta}}{\sin@@{\theta}} , x=\cos@@{\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyV{n}@{x}:=\frac{\Jacobi{-\frac12}{\frac12}{n}@{x}}{\Jacobi{-\frac12}{\frac12}{n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyV{n}@{x} =\frac{\cos@{(n+\frac12)\theta}}{\cos@{\frac12\theta}} , x=\cos@@{\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyW{n}@{x}:=(2n+1) \frac{\Jacobi{\frac12}{-\frac12}{n}@{x}}{\Jacobi{\frac12}{-\frac12}{n}@{1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyW{n}@{x} =\frac{\sin@{(n+\frac12)\theta}}{\sin@{\frac12\theta}} , x=\cos@@{\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ChebyV{n}@{-x}=(-1)^n \ChebyW{n}@{x} }}

Legendre / Spherical

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \LegendrePoly{n}@{x}=\Jacobi{0}{0}{n}@{x}=\HyperpFq{2}{1}@@{-n,n+1}{1}{\frac{1-x}{2}} }}

Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_{-1}^1\LegendrePoly{m}@{x}\LegendrePoly{n}@{x}\,dx=\frac{2}{2n+1}\,\Kronecker{m}{n} }}

Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (2n+1)x\LegendrePoly{n}@{x}=(n+1)\LegendrePoly{n+1}@{x}+n\LegendrePoly{n-1}@{x} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicLegendrePoly{n}@@{x}{x}=\monicLegendrePoly{n+1}@@{x}{x}+\frac{n^2}{(2n-1)(2n+1)}\monicLegendrePoly{n-1}@@{x}{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \LegendrePoly{n}@{x}=\binomial{2n}{n}\frac{1}{2^n}\monicLegendrePoly{n}@@{x}{x} }}

Differential equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-x^2)y''(x)-2xy'(x)+n(n+1)y(x)=0 }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\LegendrePoly{n}@{x}}}


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \LegendrePoly{n}@{x}=\frac{(-1)^n}{2^nn!}\left(\frac{d}{dx}\right)^n\left[(1-x^2)^n\right] }}

Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^{\infty}\LegendrePoly{n}@{x}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{0}{1}@@{-}{1}{\frac{(x-1)t}{2}}\,\HyperpFq{0}{1}@@{-}{1}{\frac{(x+1)t}{2}}= \sum_{n=0}^{\infty}\frac{\LegendrePoly{n}@{x}}{(n!)^2}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \expe^{xt}\,\HyperpFq{0}{1}@@{-}{1}{\frac{(x^2-1)t^2}{4}}= \sum_{n=0}^{\infty}\frac{\LegendrePoly{n}@{x}}{n!}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \HyperpFq{2}{1}@@{\gamma,1-\gamma}{1}{\frac{1-R-t}{2}}\ \HyperpFq{2}{1}@@{\gamma,1-\gamma}{1}{\frac{1-R+t}{2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\pochhammer{1-\gamma}{n}}{(n!)^2}\LegendrePoly{n}@{x}t^n }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle R=\sqrt{1-2xt+t^2}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (1-xt)^{-\gamma}\,\HyperpFq{2}{1}@@{\frac{1}{2}\gamma,\frac{1}{2}\gamma+\frac{1}{2}}{1}{\frac{(x^2-1)t^2}{(1-xt)^2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}}{n!}\LegendrePoly{n}@{x}t^n }}

Constraint(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary


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