Jacobi

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Jacobi

Hypergeometric representation

P n ( α , β ) ⁡ ( x ) = ( α + 1 ) n n ! ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ - n , n + α + β + 1 ⁢ α + 1 ⁢ 1 - x 2 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Pochhammer-symbol 𝛼 1 𝑛 𝑛 \HyperpFq 21 @ @ 𝑛 𝑛 𝛼 𝛽 1 𝛼 1 1 𝑥 2 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right% )=\frac{{\left(\alpha+1\right)_{n}}}{n!}\ \HyperpFq{2}{1}@@{-n,n+\alpha+\beta+% 1}{\alpha+1}{\frac{1-x}{2}}}}} {\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)

∫ - 1 1 ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ P m ( α , β ) ⁡ ( x ) ⁢ P n ( α , β ) ⁡ ( x ) ⁢ 𝑑 x = 2 α + β + 1 2 ⁢ n + α + β + 1 ⁢ Γ ⁡ ( n + α + 1 ) ⁢ Γ ⁡ ( n + β + 1 ) Γ ⁡ ( n + α + β + 1 ) ⁢ n ! ⁢ δ m , n superscript subscript 1 1 superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑚 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 differential-d 𝑥 superscript 2 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 1 Euler-Gamma 𝑛 𝛼 1 Euler-Gamma 𝑛 𝛽 1 Euler-Gamma 𝑛 𝛼 𝛽 1 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{-1}^{1}(1-x)^{\alpha}(1+x)^{% \beta}P^{(\alpha,\beta)}_{m}\left(x\right)P^{(\alpha,\beta)}_{n}\left(x\right)% \,dx{}=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma\left(n+\alpha+% 1\right)\Gamma\left(n+\beta+1\right)}{\Gamma\left(n+\alpha+\beta+1\right)n!}\,% \delta_{m,n}}}} {\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} }
∫ 1 ∞ ( x + 1 ) α ⁢ ( x - 1 ) β ⁢ P m ( α , β ) ⁡ ( - x ) ⁢ P n ( α , β ) ⁡ ( - x ) ⁢ 𝑑 x = - 2 α + β + 1 2 ⁢ n + α + β + 1 ⁢ Γ ⁡ ( - n - α - β ) ⁢ Γ ⁡ ( n + α + β + 1 ) Γ ⁡ ( - n - α ) ⁢ n ! ⁢ δ m , n superscript subscript 1 superscript 𝑥 1 𝛼 superscript 𝑥 1 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑚 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 differential-d 𝑥 superscript 2 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 1 Euler-Gamma 𝑛 𝛼 𝛽 Euler-Gamma 𝑛 𝛼 𝛽 1 Euler-Gamma 𝑛 𝛼 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\int_{1}^{\infty}(x+1)^{\alpha}(x-1)% ^{\beta}P^{(\alpha,\beta)}_{m}\left(-x\right)P^{(\alpha,\beta)}_{n}\left(-x% \right)\,dx{}=-\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma\left(-% n-\alpha-\beta\right)\Gamma\left(n+\alpha+\beta+1\right)}{\Gamma\left(-n-% \alpha\right)n!}\,\delta_{m,n}}}} {\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

x ⁢ P n ( α , β ) ⁡ ( x ) = 2 ⁢ ( n + 1 ) ⁢ ( n + α + β + 1 ) ( 2 ⁢ n + α + β + 1 ) ⁢ ( 2 ⁢ n + α + β + 2 ) ⁢ P n + 1 ( α , β ) ⁡ ( x ) + β 2 - α 2 ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 2 ) ⁢ P n ( α , β ) ⁡ ( x ) + 2 ⁢ ( n + α ) ⁢ ( n + β ) ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 1 ) ⁢ P n - 1 ( α , β ) ⁡ ( x ) 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 2 𝑛 1 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 1 2 𝑛 𝛼 𝛽 2 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 𝑥 superscript 𝛽 2 superscript 𝛼 2 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 2 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 2 𝑛 𝛼 𝑛 𝛽 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle xP^{(\alpha,\beta)}_{n}\left(x% \right)=\frac{2(n+1)(n+\alpha+\beta+1)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)% }P^{(\alpha,\beta)}_{n+1}\left(x\right){}+\frac{\beta^{2}-\alpha^{2}}{(2n+% \alpha+\beta)(2n+\alpha+\beta+2)}P^{(\alpha,\beta)}_{n}\left(x\right){}+\frac{% 2(n+\alpha)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}P^{(\alpha,\beta)}_% {n-1}\left(x\right)}}} {\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

x ⁢ P ^ n ( α , β ) ⁡ ( x ) ⁢ x = P ^ n + 1 ( α , β ) ⁡ ( x ) ⁢ x + β 2 - α 2 ( 2 ⁢ n + α + β ) ⁢ ( 2 ⁢ n + α + β + 2 ) ⁢ P ^ n ( α , β ) ⁡ ( x ) ⁢ x + 4 ⁢ n ⁢ ( n + α ) ⁢ ( n + β ) ⁢ ( n + α + β ) ( 2 ⁢ n + α + β - 1 ) ⁢ ( 2 ⁢ n + α + β ) 2 ⁢ ( 2 ⁢ n + α + β + 1 ) ⁢ P ^ n - 1 ( α , β ) ⁡ ( x ) ⁢ x 𝑥 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑥 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 1 𝑥 𝑥 superscript 𝛽 2 superscript 𝛼 2 2 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 2 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑥 4 𝑛 𝑛 𝛼 𝑛 𝛽 𝑛 𝛼 𝛽 2 𝑛 𝛼 𝛽 1 superscript 2 𝑛 𝛼 𝛽 2 2 𝑛 𝛼 𝛽 1 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle x{\widehat{P}}^{(\alpha,\beta)}_{n}% \left(x\right){x}={\widehat{P}}^{(\alpha,\beta)}_{n+1}\left(x\right){x}+\frac{% \beta^{2}-\alpha^{2}}{(2n+\alpha+\beta)(2n+\alpha+\beta+2)}{\widehat{P}}^{(% \alpha,\beta)}_{n}\left(x\right){x}{}+\frac{4n(n+\alpha)(n+\beta)(n+\alpha+% \beta)}{(2n+\alpha+\beta-1)(2n+\alpha+\beta)^{2}(2n+\alpha+\beta+1)}{\widehat{% P}}^{(\alpha,\beta)}_{n-1}\left(x\right){x}}}} {\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): absent {\displaystyle{\displaystyle{\displaystyle\;\;}}}


P n ( α , β ) ⁡ ( x ) = ( n + α + β + 1 ) n 2 n ⁢ n ! ⁢ P ^ n ( α , β ) ⁡ ( x ) ⁢ x Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑛 superscript 2 𝑛 𝑛 Jacobi-polynomial-monic-p 𝛼 𝛽 𝑛 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right% )=\frac{{\left(n+\alpha+\beta+1\right)_{n}}}{2^{n}n!}{\widehat{P}}^{(\alpha,% \beta)}_{n}\left(x\right){x}}}} {\displaystyle \Jacobi{\alpha}{\beta}{n}@{x}=\frac{\pochhammer{n+\alpha+\beta+1}{n}}{2^nn!}\monicJacobi{\alpha}{\beta}{n}@@{x}{x} }

Differential equation

( 1 - x 2 ) ⁢ y ′′ ⁢ ( x ) + [ β - α - ( α + β + 2 ) ⁢ x ] ⁢ y ′ ⁢ ( x ) + n ⁢ ( n + α + β + 1 ) ⁢ y ⁢ ( x ) = 0 1 superscript 𝑥 2 superscript 𝑦 ′′ 𝑥 delimited-[] 𝛽 𝛼 𝛼 𝛽 2 𝑥 superscript 𝑦 ′ 𝑥 𝑛 𝑛 𝛼 𝛽 1 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})y^{\prime\prime}(x)+\left[% \beta-\alpha-(\alpha+\beta+2)x\right]y^{\prime}(x){}+n(n+\alpha+\beta+1)y(x)=0% }}} {\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): y ⁢ ( x ) = P n ( α , β ) ⁡ ( x ) 𝑦 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle y(x)=P^{(\alpha,\beta)}_{n}\left(x% \right)}}}


Forward shift operator

d d ⁢ x ⁢ P n ( α , β ) ⁡ ( x ) = n + α + β + 1 2 ⁢ P n - 1 ( α + 1 , β + 1 ) ⁡ ( x ) 𝑑 𝑑 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 𝑛 𝛼 𝛽 1 2 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}P^{(\alpha,\beta)}_{n}% \left(x\right)=\frac{n+\alpha+\beta+1}{2}P^{(\alpha+1,\beta+1)}_{n-1}\left(x% \right)}}} {\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

( 1 - x 2 ) ⁢ d d ⁢ x ⁢ P n ( α , β ) ⁡ ( x ) + [ ( β - α ) - ( α + β ) ⁢ x ] ⁢ P n ( α , β ) ⁡ ( x ) = - 2 ⁢ ( n + 1 ) ⁢ P n + 1 ( α - 1 , β - 1 ) ⁡ ( x ) 1 superscript 𝑥 2 𝑑 𝑑 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 delimited-[] 𝛽 𝛼 𝛼 𝛽 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 2 𝑛 1 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle(1-x^{2})\frac{d}{dx}P^{(\alpha,% \beta)}_{n}\left(x\right)+\left[(\beta-\alpha)-(\alpha+\beta)x\right]P^{(% \alpha,\beta)}_{n}\left(x\right){}=-2(n+1)P^{(\alpha-1,\beta-1)}_{n+1}\left(x% \right)}}} {\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} }
d d ⁢ x ⁢ [ ( 1 - x ) α ⁢ ( 1 + x ) β ⁢ P n ( α , β ) ⁡ ( x ) ] = - 2 ⁢ ( n + 1 ) ⁢ ( 1 - x ) α - 1 ⁢ ( 1 + x ) β - 1 ⁢ P n + 1 ( α - 1 , β - 1 ) ⁡ ( x ) 𝑑 𝑑 𝑥 delimited-[] superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 2 𝑛 1 superscript 1 𝑥 𝛼 1 superscript 1 𝑥 𝛽 1 Jacobi-polynomial-P 𝛼 1 𝛽 1 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\frac{d}{dx}\left[(1-x)^{\alpha}(1+x% )^{\beta}P^{(\alpha,\beta)}_{n}\left(x\right)\right]{}=-2(n+1)(1-x)^{\alpha-1}% (1+x)^{\beta-1}P^{(\alpha-1,\beta-1)}_{n+1}\left(x\right)}}} {\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

( 1 - x ) α ⁢ ( 1 + x ) β ⁢ P n ( α , β ) ⁡ ( x ) = ( - 1 ) n 2 n ⁢ n ! ⁢ ( d d ⁢ x ) n ⁢ [ ( 1 - x ) n + α ⁢ ( 1 + x ) n + β ] superscript 1 𝑥 𝛼 superscript 1 𝑥 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 1 𝑛 superscript 2 𝑛 𝑛 superscript 𝑑 𝑑 𝑥 𝑛 delimited-[] superscript 1 𝑥 𝑛 𝛼 superscript 1 𝑥 𝑛 𝛽 {\displaystyle{\displaystyle{\displaystyle(1-x)^{\alpha}(1+x)^{\beta}P^{(% \alpha,\beta)}_{n}\left(x\right)=\frac{(-1)^{n}}{2^{n}n!}\left(\frac{d}{dx}% \right)^{n}\left[(1-x)^{n+\alpha}(1+x)^{n+\beta}\right]}}} {\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

2 α + β R ⁢ ( 1 + R - t ) α ⁢ ( 1 + R + t ) β = ∑ n = 0 ∞ P n ( α , β ) ⁡ ( x ) ⁢ t n superscript 2 𝛼 𝛽 𝑅 superscript 1 𝑅 𝑡 𝛼 superscript 1 𝑅 𝑡 𝛽 superscript subscript 𝑛 0 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{2^{\alpha+\beta}}{R(1+R-t)^{% \alpha}(1+R+t)^{\beta}}=\sum_{n=0}^{\infty}P^{(\alpha,\beta)}_{n}\left(x\right% )t^{n}}}} {\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): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}


\HyperpFq ⁢ 01 ⁢ @ ⁢ @ - α + 1 ⁢ ( x - 1 ) ⁢ t 2 ⁢ \HyperpFq ⁢ 01 ⁢ @ ⁢ @ - β + 1 ⁢ ( x + 1 ) ⁢ t 2 = ∑ n = 0 ∞ P n ( α , β ) ⁡ ( x ) ( α + 1 ) n ⁢ ( β + 1 ) n ⁢ t n \HyperpFq 01 @ @ 𝛼 1 𝑥 1 𝑡 2 \HyperpFq 01 @ @ 𝛽 1 𝑥 1 𝑡 2 superscript subscript 𝑛 0 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 superscript 𝑡 𝑛 {\displaystyle{\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{P^{(\alpha,\beta)}_{n}\left(x\right)}{{\left(\alpha+1\right)_{n}}% {\left(\beta+1\right)_{n}}}t^{n}}}} {\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 }
( 1 - t ) - α - β - 1 ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ ( α + β + 1 ) , 1 2 ⁢ ( α + β + 2 ) ⁢ α + 1 ⁢ 2 ⁢ ( x - 1 ) ⁢ t ( 1 - t ) 2 = ∑ n = 0 ∞ ( α + β + 1 ) n ( α + 1 ) n ⁢ P n ( α , β ) ⁡ ( x ) ⁢ t n superscript 1 𝑡 𝛼 𝛽 1 \HyperpFq 21 @ @ 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝛼 1 2 𝑥 1 𝑡 superscript 1 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛼 𝛽 1 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\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{{\left(\alpha+\beta+1% \right)_{n}}}{{\left(\alpha+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)% t^{n}}}} {\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 }
( 1 + t ) - α - β - 1 ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ ( α + β + 1 ) , 1 2 ⁢ ( α + β + 2 ) ⁢ β + 1 ⁢ 2 ⁢ ( x + 1 ) ⁢ t ( 1 + t ) 2 = ∑ n = 0 ∞ ( α + β + 1 ) n ( β + 1 ) n ⁢ P n ( α , β ) ⁡ ( x ) ⁢ t n superscript 1 𝑡 𝛼 𝛽 1 \HyperpFq 21 @ @ 1 2 𝛼 𝛽 1 1 2 𝛼 𝛽 2 𝛽 1 2 𝑥 1 𝑡 superscript 1 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛼 𝛽 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\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{{\left(\alpha+\beta+1\right)_% {n}}}{{\left(\beta+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)t^{n}}}} {\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 }
\HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , α + β + 1 - γ ⁢ α + 1 ⁢ 1 - R - t 2 ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ γ , α + β + 1 - γ ⁢ β + 1 ⁢ 1 - R + t 2 = ∑ n = 0 ∞ ( γ ) n ⁢ ( α + β + 1 - γ ) n ( α + 1 ) n ⁢ ( β + 1 ) n ⁢ P n ( α , β ) ⁡ ( x ) ⁢ t n \HyperpFq 21 @ @ 𝛾 𝛼 𝛽 1 𝛾 𝛼 1 1 𝑅 𝑡 2 \HyperpFq 21 @ @ 𝛾 𝛼 𝛽 1 𝛾 𝛽 1 1 𝑅 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 Pochhammer-symbol 𝛼 𝛽 1 𝛾 𝑛 Pochhammer-symbol 𝛼 1 𝑛 Pochhammer-symbol 𝛽 1 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 superscript 𝑡 𝑛 {\displaystyle{\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{{\left(% \gamma\right)_{n}}{\left(\alpha+\beta+1-\gamma\right)_{n}}}{{\left(\alpha+1% \right)_{n}}{\left(\beta+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)t^{% n}}}} {\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): R = 1 - 2 ⁢ x ⁢ t + t 2 𝑅 1 2 𝑥 𝑡 superscript 𝑡 2 {\displaystyle{\displaystyle{\displaystyle R=\sqrt{1-2xt+t^{2}}}}}


Limit relations

Wilson polynomial to Jacobi polynomial

lim t → ∞ ⁡ W n ⁡ ( 1 2 ⁢ ( 1 - x ) ⁢ t 2 ; 1 2 ⁢ ( α + 1 ) , 1 2 ⁢ ( α + 1 ) , 1 2 ⁢ ( β + 1 ) + i ⁢ t , 1 2 ( β + 1 ) - i t ) t 2 ⁢ n ⁢ n ! = P n ( α , β ) ⁡ ( x ) subscript → 𝑡 fragments Wilson-polynomial-W 𝑛 1 2 1 𝑥 superscript 𝑡 2 1 2 𝛼 1 1 2 𝛼 1 1 2 𝛽 1 imaginary-unit 𝑡 fragments 1 2 fragments ( β 1 imaginary-unit t ) superscript 𝑡 2 𝑛 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{W_{n}% \!\left(\frac{1}{2}(1-x)t^{2};\frac{1}{2}(\alpha+1),\frac{1}{2}(\alpha+1),% \frac{1}{2}(\beta+1)+\mathrm{i}t,\frac{1}{2}(\beta+1\right)-\mathrm{i}t)}{t^{2% n}n!}=P^{(\alpha,\beta)}_{n}\left(x\right)}}} {\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

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

lim N → ∞ ⁡ Q n ⁡ ( N ⁢ x ; α , β , N ) = P n ( α , β ) ⁡ ( 1 - 2 ⁢ x ) P n ( α , β ) ⁡ ( 1 ) subscript → 𝑁 Hahn-polynomial-Q 𝑛 𝑁 𝑥 𝛼 𝛽 𝑁 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 𝑥 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}\!% \left(Nx;\alpha,\beta,N\right)=\frac{P^{(\alpha,\beta)}_{n}\left(1-2x\right)}{% P^{(\alpha,\beta)}_{n}\left(1\right)}}}} {\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

lim β → ∞ ⁡ P n ( α , β ) ⁡ ( 1 - 2 ⁢ β - 1 ⁢ x ) = L n α ⁡ ( x ) subscript → 𝛽 Jacobi-polynomial-P 𝛼 𝛽 𝑛 1 2 superscript 𝛽 1 𝑥 generalized-Laguerre-polynomial-L 𝛼 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{\beta\rightarrow\infty}P^{(% \alpha,\beta)}_{n}\left(1-2\beta^{-1}x\right)=L^{\alpha}_{n}\left(x\right)}}} {\displaystyle \lim_{\beta\rightarrow\infty} \Jacobi{\alpha}{\beta}{n}@{1-2\beta^{-1}x}=\Laguerre[\alpha]{n}@{x} }

Jacobi polynomial to Bessel polynomial

lim α → - ∞ ⁡ P n ( α , a - α ) ⁡ ( 1 + α ⁢ x ) P n ( α , a - α ) ⁡ ( 1 ) = y n ⁡ ( x ; a ) subscript → 𝛼 Jacobi-polynomial-P 𝛼 𝑎 𝛼 𝑛 1 𝛼 𝑥 Jacobi-polynomial-P 𝛼 𝑎 𝛼 𝑛 1 Bessel-polynomial-y 𝑛 𝑥 𝑎 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow-\infty}\frac% {P^{(\alpha,a-\alpha)}_{n}\left(1+\alpha x\right)}{P^{(\alpha,a-\alpha)}_{n}% \left(1\right)}=y_{n}\!\left(x;a\right)}}} {\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

lim α → ∞ ⁡ α - 1 2 ⁢ n ⁢ P n ( α , α ) ⁡ ( α - 1 2 ⁢ x ) = H n ⁡ ( x ) 2 n ⁢ n ! subscript → 𝛼 superscript 𝛼 1 2 𝑛 Jacobi-polynomial-P 𝛼 𝛼 𝑛 superscript 𝛼 1 2 𝑥 Hermite-polynomial-H 𝑛 𝑥 superscript 2 𝑛 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{\alpha\rightarrow\infty}\alpha% ^{-\frac{1}{2}n}P^{(\alpha,\alpha)}_{n}\left(\alpha^{-\frac{1}{2}}x\right)=% \frac{H_{n}\left(x\right)}{2^{n}n!}}}} {\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

P n ( α , β ) ⁡ ( x ) = 1 n ! ⁢ ∑ k = 0 n ( - n ) k k ! ⁢ ( n + α + β + 1 ) k ⁢ ( α + k + 1 ) n - k ⁢ ( 1 - x 2 ) k Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 1 𝑛 superscript subscript 𝑘 0 𝑛 Pochhammer-symbol 𝑛 𝑘 𝑘 Pochhammer-symbol 𝑛 𝛼 𝛽 1 𝑘 Pochhammer-symbol 𝛼 𝑘 1 𝑛 𝑘 superscript 1 𝑥 2 𝑘 {\displaystyle{\displaystyle{\displaystyle P^{(\alpha,\beta)}_{n}\left(x\right% )=\frac{1}{n!}\sum_{k=0}^{n}\frac{{\left(-n\right)_{k}}}{k!}{\left(n+\alpha+% \beta+1\right)_{k}}{\left(\alpha+k+1\right)_{n-k}}\left(\frac{1-x}{2}\right)^{% k}}}} {\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 }
( β ) n n ! M n ⁡ ( x ; β , c ) = P n ( β - 1 , - n - β - x ) ⁡ ( ( 2 - c ) c - 1 ) fragments Pochhammer-symbol 𝛽 𝑛 𝑛 Meixner-polynomial-M 𝑛 𝑥 𝛽 𝑐 Jacobi-polynomial-P 𝛽 1 𝑛 𝛽 𝑥 𝑛 fragments ( 2 c superscript 𝑐 1 ) {\displaystyle{\displaystyle{\displaystyle\frac{{\left(\beta\right)_{n}}}{n!}M% _{n}\!\left(x;\beta,c\right)=P^{(\beta-1,-n-\beta-x)}_{n}\left((2-c\right)c^{-% 1})}}} {\displaystyle \frac{\pochhammer{\beta}{n}}{n!}\Meixner{n}@{x}{\beta}{c}=\Jacobi{\beta-1}{-n-\beta-x}{n}@{(2-c}c^{-1}) }
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} }
C 2 ⁢ n λ ⁡ ( x ) = ( λ ) n ( 1 2 ) n ⁢ P n ( λ - 1 2 , - 1 2 ) ⁡ ( 2 ⁢ x 2 - 1 ) ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 𝑥 Pochhammer-symbol 𝜆 𝑛 Pochhammer-symbol 1 2 𝑛 Jacobi-polynomial-P 𝜆 1 2 1 2 𝑛 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{2n}\left(x\right)=% \frac{{\left(\lambda\right)_{n}}}{{\left(\frac{1}{2}\right)_{n}}}P^{(\lambda-% \frac{1}{2},-\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}} {\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} }
C 2 ⁢ n + 1 λ ⁡ ( x ) = ( λ ) n + 1 ( 1 2 ) n + 1 ⁢ x ⁢ P n ( λ - 1 2 , 1 2 ) ⁡ ( 2 ⁢ x 2 - 1 ) ultraspherical-Gegenbauer-polynomial 𝜆 2 𝑛 1 𝑥 Pochhammer-symbol 𝜆 𝑛 1 Pochhammer-symbol 1 2 𝑛 1 𝑥 Jacobi-polynomial-P 𝜆 1 2 1 2 𝑛 2 superscript 𝑥 2 1 {\displaystyle{\displaystyle{\displaystyle C^{\lambda}_{2n+1}\left(x\right)=% \frac{{\left(\lambda\right)_{n+1}}}{{\left(\frac{1}{2}\right)_{n+1}}}xP^{(% \lambda-\frac{1}{2},\frac{1}{2})}_{n}\left(2x^{2}-1\right)}}} {\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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