P n â¡ ( 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 ð ð¥ ð ð superscript 2 imaginary-unit ð Pochhammer-symbol ð imaginary-unit ð ð Pochhammer-symbol ð 2 ð 1 ð \HyperpFq 21 @ @ ð ð 2 ð 1 ð imaginary-unit ð 1 imaginary-unit ð¥ 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}} }
P n â¡ ( 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 ð ð¥ ð ð superscript ð¥ imaginary-unit ð \HyperpFq 21 @ @ ð ð 1 ð imaginary-unit ð 2 ð 2 2 ð 2 1 imaginary-unit ð¥ {\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}} }
1 2 â¢ Ï â¢ â« - â â ( 1 + x 2 ) - N - 1 ⢠e 2 ⢠ν ⢠arctan â¡ x ⢠P m â¡ ( x ; ν , N ) ⢠P n â¡ ( x ; ν , N ) ⢠ð x = Î â¡ ( 2 ⢠N + 1 - 2 ⢠n ) ⢠Π⡠( 2 ⢠N + 2 - 2 ⢠n ) ⢠2 2 ⢠n - 2 ⢠N - 1 ⢠n ! Î â¡ ( 2 ⢠N + 2 - n ) ⢠| Î â¡ ( N + 1 - n + i ⢠ν ) | 2 ⢠δ m , n 1 2 superscript subscript superscript 1 superscript ð¥ 2 ð 1 2 ð ð¥ pseudo-Jacobi-polynomial ð ð¥ ð ð pseudo-Jacobi-polynomial ð ð¥ ð ð differential-d ð¥ Euler-Gamma 2 ð 1 2 ð Euler-Gamma 2 ð 2 2 ð superscript 2 2 ð 2 ð 1 ð Euler-Gamma 2 ð 2 ð superscript Euler-Gamma ð 1 ð imaginary-unit ð 2 Kronecker-delta ð ð {\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} }
x ⢠P n â¡ ( x ; ν , N ) = P n + 1 â¡ ( x ; ν , N ) + ( N + 1 ) ⢠ν ( n - N - 1 ) ⢠( n - N ) ⢠P n â¡ ( 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 â¡ ( x ; ν , N ) ð¥ pseudo-Jacobi-polynomial ð ð¥ ð ð pseudo-Jacobi-polynomial ð 1 ð¥ ð ð ð 1 ð ð ð 1 ð ð pseudo-Jacobi-polynomial ð ð¥ ð ð ð ð 2 ð 2 2 ð 2 ð 3 superscript ð ð 1 2 2 ð 2 ð 1 ð ð 1 imaginary-unit ð ð ð 1 imaginary-unit ð pseudo-Jacobi-polynomial ð 1 ð¥ ð ð {\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} }
x ⢠P ^ n â¡ ( x ) = P ^ n + 1 â¡ ( x ) + ( N + 1 ) ⢠ν ( n - N - 1 ) ⢠( n - N ) ⢠P ^ n â¡ ( 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 â¡ ( x ) ð¥ pseudo-Jacobi-polynomial-monic-p ð ð¥ ð ð pseudo-Jacobi-polynomial-monic-p ð 1 ð¥ ð ð ð 1 ð ð ð 1 ð ð pseudo-Jacobi-polynomial-monic-p ð ð¥ ð ð ð ð 2 ð 2 ð ð 1 imaginary-unit ð ð ð 1 imaginary-unit ð 2 ð 2 ð 3 superscript ð ð 1 2 2 ð 2 ð 1 pseudo-Jacobi-polynomial-monic-p ð 1 ð¥ ð ð {\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 â¡ ( x ; ν , N ) = P ^ n â¡ ( x ) pseudo-Jacobi-polynomial ð ð¥ ð ð pseudo-Jacobi-polynomial-monic-p ð ð¥ ð ð {\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} }
( 1 + x 2 ) ⢠y â²â² ⢠( x ) + 2 ⢠( ν - N ⢠x ) ⢠y Ⲡ⢠( x ) - n ⢠( n - 2 ⢠N - 1 ) ⢠y ⢠( x ) = 0 1 superscript ð¥ 2 superscript ð¦ â²â² ð¥ 2 ð ð ð¥ superscript ð¦ â² ð¥ ð ð 2 ð 1 ð¦ ð¥ 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 }
d d ⢠x ⢠P n â¡ ( x ; ν , N ) = n ⢠P n - 1 â¡ ( x ; ν , N - 1 ) ð ð ð¥ pseudo-Jacobi-polynomial ð ð¥ ð ð ð pseudo-Jacobi-polynomial ð 1 ð¥ ð ð 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} }
( 1 + x 2 ) ⢠d d ⢠x ⢠P n â¡ ( x ; ν , N ) + 2 ⢠[ ν - ( N + 1 ) ⢠x ] ⢠P n â¡ ( x ; ν , N ) = ( n - 2 ⢠N - 2 ) ⢠P n + 1 â¡ ( x ; ν , N + 1 ) 1 superscript ð¥ 2 ð ð ð¥ pseudo-Jacobi-polynomial ð ð¥ ð ð 2 delimited-[] ð ð 1 ð¥ pseudo-Jacobi-polynomial ð ð¥ ð ð ð 2 ð 2 pseudo-Jacobi-polynomial ð 1 ð¥ ð ð 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 â¡ x ⢠P n â¡ ( x ; ν , N ) ] = ( n - 2 ⢠N - 2 ) ⢠( 1 + x 2 ) - N - 2 ⢠e 2 ⢠ν ⢠arctan â¡ x ⢠P n + 1 â¡ ( x ; ν , N + 1 ) ð ð ð¥ delimited-[] superscript 1 superscript ð¥ 2 ð 1 2 ð ð¥ pseudo-Jacobi-polynomial ð ð¥ ð ð ð 2 ð 2 superscript 1 superscript ð¥ 2 ð 2 2 ð ð¥ pseudo-Jacobi-polynomial ð 1 ð¥ ð ð 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} }
P n â¡ ( x ; ν , N ) = ( 1 + x 2 ) N + 1 ⢠e - 2 ⢠ν ⢠arctan â¡ x ( n - 2 ⢠N - 1 ) n ⢠( d d ⢠x ) n ⢠[ ( 1 + x 2 ) n - N - 1 ⢠e 2 ⢠ν ⢠arctan â¡ x ] pseudo-Jacobi-polynomial ð ð¥ ð ð superscript 1 superscript ð¥ 2 ð 1 2 ð ð¥ Pochhammer-symbol ð 2 ð 1 ð superscript ð ð ð¥ ð delimited-[] superscript 1 superscript ð¥ 2 ð ð 1 2 ð ð¥ {\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] }
[ \HyperpFq 01 @ @ - - N + i ν ( x + i ) t \HyperpFq 01 @ @ - - N - i ν ( x - i ) t ] N = â n = 0 N ( n - 2 ⢠N - 1 ) n ( - N + i ⢠ν ) n ⢠( - N - i ⢠ν ) n ⢠n ! P n â¡ ( 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 ] ð superscript subscript ð 0 ð Pochhammer-symbol ð 2 ð 1 ð Pochhammer-symbol ð imaginary-unit ð ð Pochhammer-symbol ð imaginary-unit ð ð ð pseudo-Jacobi-polynomial ð ð¥ ð ð superscript ð¡ ð {\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 }
lim t â â â¡ \ctsHahn ⢠n ⢠@ ⢠x ⢠t ⢠1 2 ⢠( - N + \iunit ⢠ν - 2 ⢠t ) ⢠1 2 ⢠( - N - \iunit ⢠ν + 2 ⢠t ) ⢠1 2 ⢠( - N + \iunit ⢠ν - 2 ⢠t ) ⢠1 2 ⢠( - N - \iunit ⢠ν + 2 ⢠t ) t n = \pochhammer ⢠n - 2 ⢠N - 1 ⢠n n ! ⢠\pseudoJacobi ⢠n ⢠@ ⢠x ⢠ν ⢠N subscript â ð¡ \ctsHahn ð @ ð¥ ð¡ 1 2 ð \iunit ð 2 ð¡ 1 2 ð \iunit ð 2 ð¡ 1 2 ð \iunit ð 2 ð¡ 1 2 ð \iunit ð 2 ð¡ superscript ð¡ ð \pochhammer ð 2 ð 1 ð ð \pseudoJacobi ð @ ð¥ ð ð {\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}}}} {\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} }
( - N + i ⢠ν ) n ( - N + i ⢠ν ) k = ( - N + i ⢠ν + k ) n - k Pochhammer-symbol ð imaginary-unit ð ð Pochhammer-symbol ð imaginary-unit ð ð Pochhammer-symbol ð imaginary-unit ð ð ð ð {\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 â¡ x = ( 1 + i ⢠x ) - N - 1 - i ⢠ν ⢠( 1 - i ⢠x ) - N - 1 + i ⢠ν superscript 1 superscript ð¥ 2 ð 1 2 ð ð¥ superscript 1 imaginary-unit ð¥ ð 1 imaginary-unit ð superscript 1 imaginary-unit ð¥ ð 1 imaginary-unit ð {\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 â¡ ( x ; ν , N ) = ( - 2 ⢠i ) n ⢠n ! ( n - 2 ⢠N - 1 ) n ⢠P n ( - N - 1 + i ⢠ν , - N - 1 - i ⢠ν ) â¡ ( i ⢠x ) pseudo-Jacobi-polynomial ð ð¥ ð ð superscript 2 imaginary-unit ð ð Pochhammer-symbol ð 2 ð 1 ð Jacobi-polynomial-P ð 1 imaginary-unit ð ð 1 imaginary-unit ð ð imaginary-unit ð¥ {\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 â â â¡ P n â¡ ( ν ⢠x ; ν , N ) ν n = 2 n ( n - 2 ⢠N - 1 ) n ⢠y n â¡ ( x ; - 2 ⢠N - 2 ) subscript â ð pseudo-Jacobi-polynomial ð ð ð¥ ð ð superscript ð ð superscript 2 ð Pochhammer-symbol ð 2 ð 1 ð Bessel-polynomial-y ð ð¥ 2 ð 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} }