# Pseudo Jacobi

## Hypergeometric representation

$\displaystyle {\displaystyle \pseudoJacobi{n}@{x}{\nu}{N}=\frac{(-2\iunit)^n\pochhammer{-N+\iunit\nu}{n}}{\pochhammer{n-2N-1}{n}}\,\HyperpFq{2}{1}@@{-n,n-2N-1}{-N+\iunit\nu}{\frac{1-\iunit x}{2}} }$

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

$\displaystyle {\displaystyle \pseudoJacobi{n}@{x}{\nu}{N}=(x+\iunit)^n\,\HyperpFq{2}{1}@@{-n,N+1-n-\iunit\nu}{2N+2-2n}{\frac{2}{1-\iunit x}} }$

## Orthogonality relation(s)

$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_{-\infty}^{\infty}(1+x^2)^{-N-1}\expe^{2\nu\atan@@{x}}\pseudoJacobi{m}@{x}{\nu}{N}\pseudoJacobi{n}@{x}{\nu}{N}\,dx {}=\frac{\EulerGamma@{2N+1-2n}\EulerGamma@{2N+2-2n}2^{2n-2N-1}n!}{\EulerGamma@{2N+2-n}\left|\EulerGamma@{N+1-n+\iunit\nu}\right|^2}\,\Kronecker{m}{n} }$

## Recurrence relation

$\displaystyle {\displaystyle x\pseudoJacobi{n}@{x}{\nu}{N}=\pseudoJacobi{n+1}@{x}{\nu}{N}+\frac{(N+1)\nu}{(n-N-1)(n-N)}\pseudoJacobi{n}@{x}{\nu}{N} {}-\frac{n(n-2N-2)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)} {}(n-N-1-\iunit\nu)(n-N-1+\iunit\nu)\pseudoJacobi{n-1}@{x}{\nu}{N} }$

## Monic recurrence relation

$\displaystyle {\displaystyle x\monicpseudoJacobi{n}@@{x}{\nu}{N}=\monicpseudoJacobi{n+1}@@{x}{\nu}{N}+\frac{(N+1)\nu}{(n-N-1)(n-N)}\monicpseudoJacobi{n}@@{x}{\nu}{N} {}-\frac{n(n-2N-2)(n-N-1-\iunit\nu)(n-N-1+\iunit\nu)}{(2n-2N-3)(n-N-1)^2(2n-2N-1)}\monicpseudoJacobi{n-1}@@{x}{\nu}{N} }$
$\displaystyle {\displaystyle \pseudoJacobi{n}@{x}{\nu}{N}=\monicpseudoJacobi{n}@@{x}{\nu}{N} }$

## Differential equation

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

Substitution(s): $\displaystyle {\displaystyle y(x)=\pseudoJacobi{n}@{x}{\nu}{N}}$

## Forward shift operator

$\displaystyle {\displaystyle \frac{d}{dx}\pseudoJacobi{n}@{x}{\nu}{N}=n\pseudoJacobi{n-1}@{x}{\nu}{N-1} }$

## Backward shift operator

$\displaystyle {\displaystyle (1+x^2)\frac{d}{dx}\pseudoJacobi{n}@{x}{\nu}{N}+2\left[\nu-(N+1)x\right]\pseudoJacobi{n}@{x}{\nu}{N} {}=(n-2N-2)\pseudoJacobi{n+1}@{x}{\nu}{N+1} }$
$\displaystyle {\displaystyle \frac{d}{dx}\left[(1+x^2)^{-N-1}\expe^{2\nu\atan@@{x}}\pseudoJacobi{n}@{x}{\nu}{N}\right] {}=(n-2N-2)(1+x^2)^{-N-2}\expe^{2\nu\atan@@{x}}\pseudoJacobi{n+1}@{x}{\nu}{N+1} }$

## Rodrigues-type formula

$\displaystyle {\displaystyle \pseudoJacobi{n}@{x}{\nu}{N}=\frac{(1+x^2)^{N+1}\expe^{-2\nu\atan@@{x}}}{\pochhammer{n-2N-1}{n}} \left(\frac{d}{dx}\right)^n\left[(1+x^2)^{n-N-1}\expe^{2\nu\atan@@{x}}\right] }$

## Generating function

$\displaystyle {\displaystyle \left[\HyperpFq{0}{1}@@{-}{-N+\iunit\nu}{(x+\iunit)t}\,\HyperpFq{0}{1}@@{-}{-N-\iunit\nu}{(x-\iunit)t}\right]_N {}=\sum_{n=0}^N\frac{\pochhammer{n-2N-1}{n}}{\pochhammer{-N+\iunit\nu}{n}\pochhammer{-N-\iunit\nu}{n}n!}\pseudoJacobi{n}@{x}{\nu}{N}t^n }$

## Limit relation

### Continuous Hahn polynomial to Pseudo Jacobi polynomial

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

## Remarks

$\displaystyle {\displaystyle \frac{\pochhammer{-N+\iunit\nu}{n}}{\pochhammer{-N+\iunit\nu}{k}}=\pochhammer{-N+\iunit\nu+k}{n-k} }$
$\displaystyle {\displaystyle (1+x^2)^{-N-1}\expe^{2\nu\atan@@{x}}=(1+\iunit x)^{-N-1-\iunit\nu}(1-\iunit x)^{-N-1+\iunit\nu} }$
$\displaystyle {\displaystyle \pseudoJacobi{n}@{x}{\nu}{N}=\frac{(-2\iunit)^nn!}{\pochhammer{n-2N-1}{n}}\Jacobi{-N-1+\iunit\nu}{-N-1-\iunit\nu}{n}@{\iunit x} }$
$\displaystyle {\displaystyle \lim\limits_{\nu\rightarrow\infty}\frac{\pseudoJacobi{n}@{\nu x}{\nu}{N}}{\nu^n} =\frac{2^n}{\pochhammer{n-2N-1}{n}}\BesselPoly{n}@{x}{-2N-2} }$