Pseudo Jacobi

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Pseudo Jacobi

Hypergeometric representation

P n ⁡ ( x ; ν , N ) = ( - 2 ⁢ i ) n ⁢ ( - N + i ⁢ ν ) n ( n - 2 ⁢ N - 1 ) n ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , n - 2 ⁢ N - 1 - N + i ⁢ ν ⁢ 1 - i ⁢ x 2 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 superscript 2 imaginary-unit 𝑛 Pochhammer-symbol 𝑁 imaginary-unit 𝜈 𝑛 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 \HyperpFq 21 @ @ 𝑛 𝑛 2 𝑁 1 𝑁 imaginary-unit 𝜈 1 imaginary-unit 𝑥 2 {\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}}}}} {\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): n = 0 , 1 , 2 , … , N 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}


P n ⁡ ( x ; ν , N ) = ( x + i ) n ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , N + 1 - n - i ⁢ ν ⁢ 2 ⁢ N + 2 - 2 ⁢ n ⁢ 2 1 - i ⁢ x pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 superscript 𝑥 imaginary-unit 𝑛 \HyperpFq 21 @ @ 𝑛 𝑁 1 𝑛 imaginary-unit 𝜈 2 𝑁 2 2 𝑛 2 1 imaginary-unit 𝑥 {\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}}}}} {\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)

1 2 ⁢ π ⁢ ∫ - ∞ ∞ ( 1 + x 2 ) - N - 1 ⁢ e 2 ⁢ ν ⁢ arctan ⁡ x ⁢ P m ⁡ ( x ; ν , N ) ⁢ P n ⁡ ( x ; ν , N ) ⁢ 𝑑 x = Γ ⁡ ( 2 ⁢ N + 1 - 2 ⁢ n ) ⁢ Γ ⁡ ( 2 ⁢ N + 2 - 2 ⁢ n ) ⁢ 2 2 ⁢ n - 2 ⁢ N - 1 ⁢ n ! Γ ⁡ ( 2 ⁢ N + 2 - n ) ⁢ | Γ ⁡ ( N + 1 - n + i ⁢ ν ) | 2 ⁢ δ m , n 1 2 superscript subscript superscript 1 superscript 𝑥 2 𝑁 1 2 𝜈 𝑥 pseudo-Jacobi-polynomial 𝑚 𝑥 𝜈 𝑁 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 differential-d 𝑥 Euler-Gamma 2 𝑁 1 2 𝑛 Euler-Gamma 2 𝑁 2 2 𝑛 superscript 2 2 𝑛 2 𝑁 1 𝑛 Euler-Gamma 2 𝑁 2 𝑛 superscript Euler-Gamma 𝑁 1 𝑛 imaginary-unit 𝜈 2 Kronecker-delta 𝑚 𝑛 {\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}}}} {\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

x ⁢ P n ⁡ ( x ; ν , N ) = P n + 1 ⁡ ( x ; ν , N ) + ( N + 1 ) ⁢ ν ( n - N - 1 ) ⁢ ( n - N ) ⁢ P n ⁡ ( x ; ν , N ) - n ⁢ ( n - 2 ⁢ N - 2 ) ( 2 ⁢ n - 2 ⁢ N - 3 ) ⁢ ( n - N - 1 ) 2 ⁢ ( 2 ⁢ n - 2 ⁢ N - 1 ) ⁢ ( n - N - 1 - i ⁢ ν ) ⁢ ( n - N - 1 + i ⁢ ν ) ⁢ P n - 1 ⁡ ( x ; ν , N ) 𝑥 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 pseudo-Jacobi-polynomial 𝑛 1 𝑥 𝜈 𝑁 𝑁 1 𝜈 𝑛 𝑁 1 𝑛 𝑁 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 𝑛 𝑛 2 𝑁 2 2 𝑛 2 𝑁 3 superscript 𝑛 𝑁 1 2 2 𝑛 2 𝑁 1 𝑛 𝑁 1 imaginary-unit 𝜈 𝑛 𝑁 1 imaginary-unit 𝜈 pseudo-Jacobi-polynomial 𝑛 1 𝑥 𝜈 𝑁 {\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)}}} {\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

x ⁢ P ^ n ⁡ ( x ) = P ^ n + 1 ⁡ ( x ) + ( N + 1 ) ⁢ ν ( n - N - 1 ) ⁢ ( n - N ) ⁢ P ^ n ⁡ ( x ) - n ⁢ ( n - 2 ⁢ N - 2 ) ⁢ ( n - N - 1 - i ⁢ ν ) ⁢ ( n - N - 1 + i ⁢ ν ) ( 2 ⁢ n - 2 ⁢ N - 3 ) ⁢ ( n - N - 1 ) 2 ⁢ ( 2 ⁢ n - 2 ⁢ N - 1 ) ⁢ P ^ n - 1 ⁡ ( x ) 𝑥 pseudo-Jacobi-polynomial-monic-p 𝑛 𝑥 𝜈 𝑁 pseudo-Jacobi-polynomial-monic-p 𝑛 1 𝑥 𝜈 𝑁 𝑁 1 𝜈 𝑛 𝑁 1 𝑛 𝑁 pseudo-Jacobi-polynomial-monic-p 𝑛 𝑥 𝜈 𝑁 𝑛 𝑛 2 𝑁 2 𝑛 𝑁 1 imaginary-unit 𝜈 𝑛 𝑁 1 imaginary-unit 𝜈 2 𝑛 2 𝑁 3 superscript 𝑛 𝑁 1 2 2 𝑛 2 𝑁 1 pseudo-Jacobi-polynomial-monic-p 𝑛 1 𝑥 𝜈 𝑁 {\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 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} }
P n ⁡ ( x ; ν , N ) = P ^ n ⁡ ( x ) pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 pseudo-Jacobi-polynomial-monic-p 𝑛 𝑥 𝜈 𝑁 {\displaystyle{\displaystyle{\displaystyle P_{n}\!\left(x;\nu,N\right)={% \widehat{P}}_{n}\!\left(x\right)}}} {\displaystyle \pseudoJacobi{n}@{x}{\nu}{N}=\monicpseudoJacobi{n}@@{x}{\nu}{N} }

Differential equation

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

Substitution(s): y ⁢ ( x ) = P n ⁡ ( x ; ν , N ) 𝑦 𝑥 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 {\displaystyle{\displaystyle{\displaystyle y(x)=P_{n}\!\left(x;\nu,N\right)}}}


Forward shift operator

d d ⁢ x ⁢ P n ⁡ ( x ; ν , N ) = n ⁢ P n - 1 ⁡ ( x ; ν , N - 1 ) 𝑑 𝑑 𝑥 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 𝑛 pseudo-Jacobi-polynomial 𝑛 1 𝑥 𝜈 𝑁 1 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}P_{n}\!\left(x;\nu,N% \right)=nP_{n-1}\!\left(x;\nu,N-1\right)}}} {\displaystyle \frac{d}{dx}\pseudoJacobi{n}@{x}{\nu}{N}=n\pseudoJacobi{n-1}@{x}{\nu}{N-1} }

Backward shift operator

( 1 + x 2 ) ⁢ d d ⁢ x ⁢ P n ⁡ ( x ; ν , N ) + 2 ⁢ [ ν - ( N + 1 ) ⁢ x ] ⁢ P n ⁡ ( x ; ν , N ) = ( n - 2 ⁢ N - 2 ) ⁢ P n + 1 ⁡ ( x ; ν , N + 1 ) 1 superscript 𝑥 2 𝑑 𝑑 𝑥 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 2 delimited-[] 𝜈 𝑁 1 𝑥 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 𝑛 2 𝑁 2 pseudo-Jacobi-polynomial 𝑛 1 𝑥 𝜈 𝑁 1 {\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 (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} }
d d ⁢ x ⁢ [ ( 1 + x 2 ) - N - 1 ⁢ e 2 ⁢ ν ⁢ arctan ⁡ x ⁢ P n ⁡ ( x ; ν , N ) ] = ( n - 2 ⁢ N - 2 ) ⁢ ( 1 + x 2 ) - N - 2 ⁢ e 2 ⁢ ν ⁢ arctan ⁡ x ⁢ P n + 1 ⁡ ( x ; ν , N + 1 ) 𝑑 𝑑 𝑥 delimited-[] superscript 1 superscript 𝑥 2 𝑁 1 2 𝜈 𝑥 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 𝑛 2 𝑁 2 superscript 1 superscript 𝑥 2 𝑁 2 2 𝜈 𝑥 pseudo-Jacobi-polynomial 𝑛 1 𝑥 𝜈 𝑁 1 {\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)}}} {\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

P n ⁡ ( x ; ν , N ) = ( 1 + x 2 ) N + 1 ⁢ e - 2 ⁢ ν ⁢ arctan ⁡ x ( n - 2 ⁢ N - 1 ) n ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 + x 2 ) n - N - 1 ⁢ e 2 ⁢ ν ⁢ arctan ⁡ x ] pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 superscript 1 superscript 𝑥 2 𝑁 1 2 𝜈 𝑥 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 superscript 𝑥 2 𝑛 𝑁 1 2 𝜈 𝑥 {\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]}}} {\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

[ \HyperpFq 01 @ @ - - N + i ν ( x + i ) t \HyperpFq 01 @ @ - - N - i ν ( x - i ) t ] N = ∑ n = 0 N ( n - 2 ⁢ N - 1 ) n ( - N + i ⁢ ν ) n ⁢ ( - N - i ⁢ ν ) n ⁢ n ! P n ⁡ ( x ; ν , N ) t n fragments subscript fragments [ \HyperpFq 01 @ @ N imaginary-unit ν fragments ( x imaginary-unit ) t \HyperpFq 01 @ @ N imaginary-unit ν fragments ( x imaginary-unit ) t ] 𝑁 superscript subscript 𝑛 0 𝑁 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 Pochhammer-symbol 𝑁 imaginary-unit 𝜈 𝑛 Pochhammer-symbol 𝑁 imaginary-unit 𝜈 𝑛 𝑛 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 superscript 𝑡 𝑛 {\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}}}} {\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

lim t → ∞ ⁡ p n ⁡ ( x ⁢ t ; 1 2 ⁢ ( - N + i ⁢ ν - 2 ⁢ t ) , 1 2 ⁢ ( - N - i ⁢ ν + 2 ⁢ t ) , 1 2 ⁢ ( - N + i ⁢ ν - 2 ⁢ t ) , 1 2 ( - N - i ν + 2 t ) ) t n = ( n - 2 ⁢ N - 1 ) n n ! ⁢ P n ⁡ ( x ; ν , N ) subscript → 𝑡 fragments continuous-Hahn-polynomial 𝑛 𝑥 𝑡 1 2 𝑁 imaginary-unit 𝜈 2 𝑡 1 2 𝑁 imaginary-unit 𝜈 2 𝑡 1 2 𝑁 imaginary-unit 𝜈 2 𝑡 fragments 1 2 fragments ( N imaginary-unit ν 2 t ) superscript 𝑡 𝑛 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 𝑛 pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{p_{n}% \!\left(xt;\frac{1}{2}(-N+\mathrm{i}\nu-2t),\frac{1}{2}(-N-\mathrm{i}\nu+2t),% \frac{1}{2}(-N+\mathrm{i}\nu-2t),\frac{1}{2}(-N-\mathrm{i}\nu+2t\right))}{t^{n% }}{}=\frac{{\left(n-2N-1\right)_{n}}}{n!}P_{n}\!\left(x;\nu,N\right)}}} {\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

( - N + i ⁢ ν ) n ( - N + i ⁢ ν ) k = ( - N + i ⁢ ν + k ) n - k Pochhammer-symbol 𝑁 imaginary-unit 𝜈 𝑛 Pochhammer-symbol 𝑁 imaginary-unit 𝜈 𝑘 Pochhammer-symbol 𝑁 imaginary-unit 𝜈 𝑘 𝑛 𝑘 {\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 \frac{\pochhammer{-N+\iunit\nu}{n}}{\pochhammer{-N+\iunit\nu}{k}}=\pochhammer{-N+\iunit\nu+k}{n-k} }
( 1 + x 2 ) - N - 1 ⁢ e 2 ⁢ ν ⁢ arctan ⁡ x = ( 1 + i ⁢ x ) - N - 1 - i ⁢ ν ⁢ ( 1 - i ⁢ x ) - N - 1 + i ⁢ ν superscript 1 superscript 𝑥 2 𝑁 1 2 𝜈 𝑥 superscript 1 imaginary-unit 𝑥 𝑁 1 imaginary-unit 𝜈 superscript 1 imaginary-unit 𝑥 𝑁 1 imaginary-unit 𝜈 {\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 (1+x^2)^{-N-1}\expe^{2\nu\atan@@{x}}=(1+\iunit x)^{-N-1-\iunit\nu}(1-\iunit x)^{-N-1+\iunit\nu} }
P n ⁡ ( x ; ν , N ) = ( - 2 ⁢ i ) n ⁢ n ! ( n - 2 ⁢ N - 1 ) n ⁢ P n ( - N - 1 + i ⁢ ν , - N - 1 - i ⁢ ν ) ⁡ ( i ⁢ x ) pseudo-Jacobi-polynomial 𝑛 𝑥 𝜈 𝑁 superscript 2 imaginary-unit 𝑛 𝑛 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 Jacobi-polynomial-P 𝑁 1 imaginary-unit 𝜈 𝑁 1 imaginary-unit 𝜈 𝑛 imaginary-unit 𝑥 {\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 \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} }
lim ν → ∞ ⁡ P n ⁡ ( ν ⁢ x ; ν , N ) ν n = 2 n ( n - 2 ⁢ N - 1 ) n ⁢ y n ⁡ ( x ; - 2 ⁢ N - 2 ) subscript → 𝜈 pseudo-Jacobi-polynomial 𝑛 𝜈 𝑥 𝜈 𝑁 superscript 𝜈 𝑛 superscript 2 𝑛 Pochhammer-symbol 𝑛 2 𝑁 1 𝑛 Bessel-polynomial-y 𝑛 𝑥 2 𝑁 2 {\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)}}} {\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} }

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