Pseudo Jacobi

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Pseudo Jacobi

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Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;\nu,N\right)=\frac{(-2\mathrm{i})^{n}{\left(-N+\mathrm{i}\nu\right)_{n}}}{{\left(n-2N-1\right)_{n}}}\,\HyperpFq{2}{1}@@{-n,n-2N-1}{-N+\mathrm{i}\nu}{\frac{1-\mathrm{i}x}{2}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}$

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;\nu,N\right)=(x+\mathrm{i})^{n}\,\HyperpFq{2}{1}@@{-n,N+1-n-\mathrm{i}\nu}{2N+2-2n}{\frac{2}{1-\mathrm{i}x}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}(1+x^{2})^{-N-1}{\mathrm{e}^{2\nu\operatorname{arctan}x}}P_{m}\!\left(x;\nu,N\right)P_{n}\!\left(x;\nu,N\right)\,dx{}=\frac{\Gamma\left(2N+1-2n\right)\Gamma\left(2N+2-2n\right)2^{2n-2N-1}n!}{\Gamma\left(2N+2-n\right)\left|\Gamma\left(N+1-n+\mathrm{i}\nu\right)\right|^{2}}\,\delta_{m,n}}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle xP_{n}\!\left(x;\nu,N\right)=P_{n+1}\!\left(x;\nu,N\right)+\frac{(N+1)\nu}{(n-N-1)(n-N)}P_{n}\!\left(x;\nu,N\right){}-\frac{n(n-2N-2)}{(2n-2N-3)(n-N-1)^{2}(2n-2N-1)}{}(n-N-1-\mathrm{i}\nu)(n-N-1+\mathrm{i}\nu)P_{n-1}\!\left(x;\nu,N\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}_{n}\!\left(x\right)={\widehat{P}}_{n+1}\!\left(x\right)+\frac{(N+1)\nu}{(n-N-1)(n-N)}{\widehat{P}}_{n}\!\left(x\right){}-\frac{n(n-2N-2)(n-N-1-\mathrm{i}\nu)(n-N-1+\mathrm{i}\nu)}{(2n-2N-3)(n-N-1)^{2}(2n-2N-1)}{\widehat{P}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;\nu,N\right)={\widehat{P}}_{n}\!\left(x\right)}}}$

Differential equation

${\displaystyle{\displaystyle{\displaystyle(1+x^{2})y^{\prime\prime}(x)+2\left(\nu-Nx\right)y^{\prime}(x)-n(n-2N-1)y(x)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=P_{n}\!\left(x;\nu,N\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}P_{n}\!\left(x;\nu,N\right)=nP_{n-1}\!\left(x;\nu,N-1\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle(1+x^{2})\frac{d}{dx}P_{n}\!\left(x;\nu,N\right)+2\left[\nu-(N+1)x\right]P_{n}\!\left(x;\nu,N\right){}=(n-2N-2)P_{n+1}\!\left(x;\nu,N+1\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[(1+x^{2})^{-N-1}{\mathrm{e}^{2\nu\operatorname{arctan}x}}P_{n}\!\left(x;\nu,N\right)\right]{}=(n-2N-2)(1+x^{2})^{-N-2}{\mathrm{e}^{2\nu\operatorname{arctan}x}}P_{n+1}\!\left(x;\nu,N+1\right)}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;\nu,N\right)=\frac{(1+x^{2})^{N+1}{\mathrm{e}^{-2\nu\operatorname{arctan}x}}}{{\left(n-2N-1\right)_{n}}}\left(\frac{d}{dx}\right)^{n}\left[(1+x^{2})^{n-N-1}{\mathrm{e}^{2\nu\operatorname{arctan}x}}\right]}}}$

Generating function

${\displaystyle{\displaystyle{\displaystyle\left[\HyperpFq{0}{1}@@{-}{-N+\mathrm{i}\nu}{(x+\mathrm{i})t}\,\HyperpFq{0}{1}@@{-}{-N-\mathrm{i}\nu}{(x-\mathrm{i})t}\right]_{N}{}=\sum_{n=0}^{N}\frac{{\left(n-2N-1\right)_{n}}}{{\left(-N+\mathrm{i}\nu\right)_{n}}{\left(-N-\mathrm{i}\nu\right)_{n}}n!}P_{n}\!\left(x;\nu,N\right)t^{n}}}}$

Limit relation

Continuous Hahn polynomial to Pseudo Jacobi polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{\ctsHahn{n}@{xt}{\frac{1}{2}(-N+\iunit\nu-2t)}{\frac{1}{2}(-N-\iunit\nu+2t)}{\frac{1}{2}(-N+\iunit\nu-2t)}{\frac{1}{2}(-N-\iunit\nu+2t})}{t^{n}}{}=\frac{\pochhammer{n-2N-1}{n}}{n!}\pseudoJacobi{n}@{x}{\nu}{N}}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle\frac{{\left(-N+\mathrm{i}\nu\right)_{n}}}{{\left(-N+\mathrm{i}\nu\right)_{k}}}={\left(-N+\mathrm{i}\nu+k\right)_{n-k}}}}}$
${\displaystyle{\displaystyle{\displaystyle(1+x^{2})^{-N-1}{\mathrm{e}^{2\nu\operatorname{arctan}x}}=(1+\mathrm{i}x)^{-N-1-\mathrm{i}\nu}(1-\mathrm{i}x)^{-N-1+\mathrm{i}\nu}}}}$
${\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;\nu,N\right)=\frac{(-2\mathrm{i})^{n}n!}{{\left(n-2N-1\right)_{n}}}P^{(-N-1+\mathrm{i}\nu,-N-1-\mathrm{i}\nu)}_{n}\left(\mathrm{i}x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim\limits_{\nu\rightarrow\infty}\frac{P_{n}\!\left(\nu x;\nu,N\right)}{\nu^{n}}=\frac{2^{n}}{{\left(n-2N-1\right)_{n}}}y_{n}\!\left(x;-2N-2\right)}}}$

<< Jacobi: Special cases

Pseudo Jacobi

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