Jacobi

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Jacobi

Hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Jacobi{\alpha}{\beta}{n}@{x}=\frac{\pochhammer{\alpha+1}{n}}{n!}\ \HyperpFq{2}{1}@@{-n,n+\alpha+\beta+1}{\alpha+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(1-x)^{\alpha}(1+x)^{\beta}\Jacobi{\alpha}{\beta}{m}@{x}\Jacobi{\alpha}{\beta}{n}@{x}\,dx {}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\EulerGamma@{n+\alpha+1}\EulerGamma@{n+\beta+1}}{\EulerGamma@{n+\alpha+\beta+1}n!}\,\Kronecker{m}{n} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int_1^{\infty}(x+1)^{\alpha}(x-1)^{\beta}\Jacobi{\alpha}{\beta}{m}@{-x}\Jacobi{\alpha}{\beta}{n}@{-x}\,dx {}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\EulerGamma@{-n-\alpha-\beta}\EulerGamma@{n+\alpha+\beta+1}}{\EulerGamma@{-n-\alpha}n!}\,\Kronecker{m}{n} }}

Recurrence relation

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

Monic recurrence relation

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

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


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

Generating functions

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


Limit relations

Wilson polynomial to Jacobi polynomial

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

Continuous Hahn polynomial to Jacobi polynomial

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

Hahn polynomial to Jacobi polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{N\rightarrow\infty} \Hahn{n}@{Nx}{\alpha}{\beta}{N}=\frac{\Jacobi{\alpha}{\beta}{n}@{1-2x}}{\Jacobi{\alpha}{\beta}{n}@{1}} }}

Jacobi polynomial to Laguerre polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\beta\rightarrow\infty} \Jacobi{\alpha}{\beta}{n}@{1-2\beta^{-1}x}=\Laguerre[\alpha]{n}@{x} }}

Jacobi polynomial to Bessel polynomial

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

Jacobi 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}\Jacobi{\alpha}{\alpha}{n}@{\alpha^{-\frac{1}{2}}x}=\frac{\Hermite{n}@{x}}{2^nn!} }}

Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \Jacobi{\alpha}{\beta}{n}@{x}=\frac{1}{n!}\sum_{k=0}^n\frac{\pochhammer{-n}{k}}{k!} \pochhammer{n+\alpha+\beta+1}{k}\pochhammer{\alpha+k+1}{n-k}\left(\frac{1-x}{2}\right)^k }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\pochhammer{\beta}{n}}{n!}\Meixner{n}@{x}{\beta}{c}=\Jacobi{\beta-1}{-n-\beta-x}{n}@{(2-c}c^{-1}) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \pseudoJacobi{n}@{x}{\nu}{N}=\frac{(-2\iunit)^nn!}{\pochhammer{n-2N-1}{n}}\Jacobi{-N-1+\iunit\nu}{-N-1-\iunit\nu}{n}@{\iunit x} }}
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} }}

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