Continuous Hahn

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Continuous Hahn

Hypergeometric representation

p n ⁡ ( x ; a , b , c , d ) = i n ⁢ ( a + c ) n ⁢ ( a + d ) n n ! ⁢ \HyperpFq ⁢ 32 ⁢ @ ⁢ @ - n , n + a + b + c + d - 1 , a + i ⁢ x ⁢ a + c , a + d ⁢ 1 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 imaginary-unit 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 𝑛 \HyperpFq 32 @ @ 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑎 imaginary-unit 𝑥 𝑎 𝑐 𝑎 𝑑 1 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c,d\right){}={% \mathrm{i}^{n}}\frac{{\left(a+c\right)_{n}}{\left(a+d\right)_{n}}}{n!}\,% \HyperpFq{3}{2}@@{-n,n+a+b+c+d-1,a+\mathrm{i}x}{a+c,a+d}{1}}}} {\displaystyle \ctsHahn{n}@{x}{a}{b}{c}{d} {}=\iunit^n\frac{\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{n!}\,\HyperpFq{3}{2}@@{-n,n+a+b+c+d-1,a+\iunit x}{a+c,a+d}{1} }

Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - ∞ ∞ Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c - i ⁢ x ) ⁢ Γ ⁡ ( d - i ⁢ x ) ⁢ p m ⁡ ( x ; a , b , c , d ) ⁢ p n ⁡ ( x ; a , b , c , d ) ⁢ 𝑑 x = Γ ⁡ ( n + a + c ) ⁢ Γ ⁡ ( n + a + d ) ⁢ Γ ⁡ ( n + b + c ) ⁢ Γ ⁡ ( n + b + d ) ( 2 ⁢ n + a + b + c + d - 1 ) ⁢ Γ ⁡ ( n + a + b + c + d - 1 ) ⁢ n ! ⁢ δ m , n 1 2 superscript subscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 continuous-Hahn-polynomial 𝑚 𝑥 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 differential-d 𝑥 Euler-Gamma 𝑛 𝑎 𝑐 Euler-Gamma 𝑛 𝑎 𝑑 Euler-Gamma 𝑛 𝑏 𝑐 Euler-Gamma 𝑛 𝑏 𝑑 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 Euler-Gamma 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty% }\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma\left(c% -\mathrm{i}x\right)\Gamma\left(d-\mathrm{i}x\right)p_{m}\!\left(x;a,b,c,d% \right)p_{n}\!\left(x;a,b,c,d\right)\,dx{}=\frac{\Gamma\left(n+a+c\right)% \Gamma\left(n+a+d\right)\Gamma\left(n+b+c\right)\Gamma\left(n+b+d\right)}{(2n+% a+b+c+d-1)\Gamma\left(n+a+b+c+d-1\right)n!}\,\delta_{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_{-\infty}^{\infty}\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c-\iunit x}\EulerGamma@{d-\iunit x} \ctsHahn{m}@{x}{a}{b}{c}{d}\ctsHahn{n}@{x}{a}{b}{c}{d}\,dx {}=\frac{\EulerGamma@{n+a+c}\EulerGamma@{n+a+d}\EulerGamma@{n+b+c}\EulerGamma@{n+b+d}} {(2n+a+b+c+d-1)\EulerGamma@{n+a+b+c+d-1}n!}\,\Kronecker{m}{n} }

Recurrence relation

( a + i ⁢ x ) ⁢ p ~ n ⁡ ( x ) = A n ⁢ p ~ n + 1 ⁡ ( x ) - ( A n + C n ) ⁢ p ~ n ⁡ ( x ) + C n ⁢ p ~ n - 1 ⁡ ( x ) 𝑎 imaginary-unit 𝑥 continuous-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 continuous-Hahn-polynomial-normalized-p-tilde 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 subscript 𝐶 𝑛 continuous-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐶 𝑛 continuous-Hahn-polynomial-normalized-p-tilde 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle(a+\mathrm{i}x){\tilde{p}}_{n}\!% \left(x\right)=A_{n}{\tilde{p}}_{n+1}\!\left(x\right)-\left(A_{n}+C_{n}\right)% {\tilde{p}}_{n}\!\left(x\right)+C_{n}{\tilde{p}}_{n-1}\!\left(x\right)}}} {\displaystyle (a+\iunit x)\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}=A_n\normctsHahnptilde{n+1}@@{x}{a}{b}{c}{d}-\left(A_n+C_n\right)\normctsHahnptilde{n}@@{x}{a}{b}{c}{d}+C_n\normctsHahnptilde{n-1}@@{x}{a}{b}{c}{d} }

Substitution(s): C n = n ⁢ ( n + b + c - 1 ) ⁢ ( n + b + d - 1 ) ( 2 ⁢ n + a + b + c + d - 2 ) ⁢ ( 2 ⁢ n + a + b + c + d - 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝑏 𝑐 1 𝑛 𝑏 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 2 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+b+c-1)(n+b+d-1)}{(2% n+a+b+c+d-2)(2n+a+b+c+d-1)}}}} &
A n = - ( n + a + b + c + d - 1 ) ⁢ ( n + a + c ) ⁢ ( n + a + d ) ( 2 ⁢ n + a + b + c + d - 1 ) ⁢ ( 2 ⁢ n + a + b + c + d ) subscript 𝐴 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑎 𝑐 𝑛 𝑎 𝑑 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle A_{n}=-\frac{(n+a+b+c+d-1)(n+a+c)(n% +a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}}}


p ~ n ⁡ ( x ) := p ~ n ⁡ ( x ; a , b , c , d ) = n ! i n ⁢ ( a + c ) n ⁢ ( a + d ) n ⁢ p n ⁡ ( x ; a , b , c , d ) assign continuous-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝑛 imaginary-unit 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x\right):={% \tilde{p}}_{n}\!\left(x;a,b,c,d\right)=\frac{n!}{{\mathrm{i}^{n}}{\left(a+c% \right)_{n}}{\left(a+d\right)_{n}}}p_{n}\!\left(x;a,b,c,d\right)}}} {\displaystyle \normctsHahnptilde{n}@@{x}{a}{b}{c}{d}:=\normctsHahnptilde{n}@{x}{a}{b}{c}{d}=\frac{n!}{\iunit^n\pochhammer{a+c}{n}\pochhammer{a+d}{n}}\ctsHahn{n}@{x}{a}{b}{c}{d} }

Monic recurrence relation

x ⁢ p ^ n ⁡ ( x ) = p ^ n + 1 ⁡ ( x ) + i ⁢ ( A n + C n + a ) ⁢ p ^ n ⁡ ( x ) - A n - 1 ⁢ C n ⁢ p ^ n - 1 ⁡ ( x ) 𝑥 continuous-Hahn-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 imaginary-unit subscript 𝐴 𝑛 subscript 𝐶 𝑛 𝑎 continuous-Hahn-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 continuous-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)=% {\widehat{p}}_{n+1}\!\left(x\right)+\mathrm{i}(A_{n}+C_{n}+a){\widehat{p}}_{n}% \!\left(x\right)-A_{n-1}C_{n}{\widehat{p}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicctsHahn{n}@@{x}{a}{b}{c}{d}=\monicctsHahn{n+1}@@{x}{a}{b}{c}{d}+\iunit(A_n+C_n+a)\monicctsHahn{n}@@{x}{a}{b}{c}{d}-A_{n-1}C_n\monicctsHahn{n-1}@@{x}{a}{b}{c}{d} }

Substitution(s): C n = n ⁢ ( n + b + c - 1 ) ⁢ ( n + b + d - 1 ) ( 2 ⁢ n + a + b + c + d - 2 ) ⁢ ( 2 ⁢ n + a + b + c + d - 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝑏 𝑐 1 𝑛 𝑏 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 2 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=\frac{n(n+b+c-1)(n+b+d-1)}{(2% n+a+b+c+d-2)(2n+a+b+c+d-1)}}}} &
A n = - ( n + a + b + c + d - 1 ) ⁢ ( n + a + c ) ⁢ ( n + a + d ) ( 2 ⁢ n + a + b + c + d - 1 ) ⁢ ( 2 ⁢ n + a + b + c + d ) subscript 𝐴 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑎 𝑐 𝑛 𝑎 𝑑 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle A_{n}=-\frac{(n+a+b+c+d-1)(n+a+c)(n% +a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}}}


p n ⁡ ( x ; a , b , c , d ) = ( n + a + b + c + d - 1 ) n n ! ⁢ p ^ n ⁡ ( x ) continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑛 continuous-Hahn-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c,d\right)=\frac% {{\left(n+a+b+c+d-1\right)_{n}}}{n!}{\widehat{p}}_{n}\!\left(x\right)}}} {\displaystyle \ctsHahn{n}@{x}{a}{b}{c}{d}=\frac{\pochhammer{n+a+b+c+d-1}{n}}{n!}\monicctsHahn{n}@@{x}{a}{b}{c}{d} }

Difference equation

n ⁢ ( n + a + b + c + d - 1 ) ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( x + i ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( x - i ) 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 imaginary-unit delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 imaginary-unit {\displaystyle{\displaystyle{\displaystyle n(n+a+b+c+d-1)y(x){}=B(x)y(x+% \mathrm{i})-\left[B(x)+D(x)\right]y(x)+D(x)y(x-\mathrm{i})}}} {\displaystyle n(n+a+b+c+d-1)y(x) {}=B(x)y(x+\iunit)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-\iunit) }

Substitution(s): D ⁢ ( x ) = ( a + i ⁢ x ) ⁢ ( b + i ⁢ x ) 𝐷 𝑥 𝑎 imaginary-unit 𝑥 𝑏 imaginary-unit 𝑥 {\displaystyle{\displaystyle{\displaystyle D(x)=(a+\mathrm{i}x)(b+\mathrm{i}x)% }}} &

B ⁢ ( x ) = ( c - i ⁢ x ) ⁢ ( d - i ⁢ x ) 𝐵 𝑥 𝑐 imaginary-unit 𝑥 𝑑 imaginary-unit 𝑥 {\displaystyle{\displaystyle{\displaystyle B(x)=(c-\mathrm{i}x)(d-\mathrm{i}x)% }}} &

y ⁢ ( x ) = p n ⁡ ( x ; a , b , c , d ) 𝑦 𝑥 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle y(x)=p_{n}\!\left(x;a,b,c,d\right)}}}


Forward shift operator

p n ⁡ ( x + 1 2 ⁢ i ; a , b , c , d ) - p n ⁡ ( x - 1 2 ⁢ i ; a , b , c , d ) = i ⁢ ( n + a + b + c + d - 1 ) ⁢ p n - 1 ⁡ ( x ; a + 1 2 , b + 1 2 , c + 1 2 , d + 1 2 ) continuous-Hahn-polynomial 𝑛 𝑥 1 2 imaginary-unit 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial 𝑛 𝑥 1 2 imaginary-unit 𝑎 𝑏 𝑐 𝑑 imaginary-unit 𝑛 𝑎 𝑏 𝑐 𝑑 1 continuous-Hahn-polynomial 𝑛 1 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x+\textstyle\frac{1}{2% }\mathrm{i};a,b,c,d\right)-p_{n}\!\left(x-\textstyle\frac{1}{2}\mathrm{i};a,b,% c,d\right){}=\mathrm{i}(n+a+b+c+d-1)p_{n-1}\!\left(x;a+\textstyle\frac{1}{2},b% +\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}\right)}}} {\displaystyle \ctsHahn{n}@{x+\textstyle\frac{1}{2}\iunit}{a}{b}{c}{d}-\ctsHahn{n}@{x-\textstyle\frac{1}{2}\iunit}{a}{b}{c}{d} {}=\iunit(n+a+b+c+d-1)\ctsHahn{n-1}@{x}{a+\textstyle\frac{1}{2}}{ b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}} }
δ ⁢ p n ⁡ ( x ; a , b , c , d ) δ ⁢ x = ( n + a + b + c + d - 1 ) ⁢ p n - 1 ⁡ ( x ; a + 1 2 , b + 1 2 , c + 1 2 , d + 1 2 ) 𝛿 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝛿 𝑥 𝑛 𝑎 𝑏 𝑐 𝑑 1 continuous-Hahn-polynomial 𝑛 1 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle\frac{\delta p_{n}\!\left(x;a,b,c,d% \right)}{\delta x}=(n+a+b+c+d-1)p_{n-1}\!\left(x;a+\textstyle\frac{1}{2},b+% \textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{1}{2}\right)}}} {\displaystyle \frac{\delta \ctsHahn{n}@{x}{a}{b}{c}{d}}{\delta x}=(n+a+b+c+d-1) \ctsHahn{n-1}@{x}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{ c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}} }

Backward shift operator

( c - 1 2 - i ⁢ x ) ⁢ ( d - 1 2 - i ⁢ x ) ⁢ p n ⁡ ( x + 1 2 ⁢ i ; a , b , c , d ) - ( a - 1 2 + i ⁢ x ) ⁢ ( b - 1 2 + i ⁢ x ) ⁢ p n ⁡ ( x - 1 2 ⁢ i ; a , b , c , d ) = n + 1 i ⁢ p n + 1 ⁡ ( x ; a - 1 2 , b - 1 2 , c - 1 2 , d - 1 2 ) 𝑐 1 2 imaginary-unit 𝑥 𝑑 1 2 imaginary-unit 𝑥 continuous-Hahn-polynomial 𝑛 𝑥 1 2 imaginary-unit 𝑎 𝑏 𝑐 𝑑 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 continuous-Hahn-polynomial 𝑛 𝑥 1 2 imaginary-unit 𝑎 𝑏 𝑐 𝑑 𝑛 1 imaginary-unit continuous-Hahn-polynomial 𝑛 1 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle(c-\textstyle\frac{1}{2}-\mathrm{i}x% )(d-\textstyle\frac{1}{2}-\mathrm{i}x)p_{n}\!\left(x+\textstyle\frac{1}{2}% \mathrm{i};a,b,c,d\right){}-(a-\textstyle\frac{1}{2}+\mathrm{i}x)(b-\textstyle% \frac{1}{2}+\mathrm{i}x)p_{n}\!\left(x-\textstyle\frac{1}{2}\mathrm{i};a,b,c,d% \right){}=\frac{n+1}{\mathrm{i}}p_{n+1}\!\left(x;a-\textstyle\frac{1}{2},b-% \textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}\right)}}} {\displaystyle (c-\textstyle\frac{1}{2}-\iunit x)(d-\textstyle\frac{1}{2}-\iunit x) \ctsHahn{n}@{x+\textstyle\frac{1}{2}\iunit}{a}{b}{c}{d} {}-(a-\textstyle\frac{1}{2}+\iunit x)(b-\textstyle\frac{1}{2}+\iunit x) \ctsHahn{n}@{x-\textstyle\frac{1}{2}\iunit}{a}{b}{c}{d} {}=\frac{n+1}{\iunit}\ctsHahn{n+1}@{x}{a-\textstyle\frac{1}{2}}{ b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\textstyle\frac{1}{2}} }
δ ⁢ [ ω ⁢ ( x ; a , b , c , d ) ⁢ p n ⁡ ( x ; a , b , c , d ) ] δ ⁢ x = - ( n + 1 ) ⁢ ω ⁢ ( x ; a - 1 2 , b - 1 2 ⁢ c - 1 2 , d - 1 2 ) ⁢ p n + 1 ⁡ ( x ; a - 1 2 , b - 1 2 , c - 1 2 , d - 1 2 ) 𝛿 delimited-[] 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 𝛿 𝑥 𝑛 1 𝜔 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 continuous-Hahn-polynomial 𝑛 1 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle\frac{\delta\left[\omega(x;a,b,c,d)p% _{n}\!\left(x;a,b,c,d\right)\right]}{\delta x}{}=-(n+1)\omega(x;a-\textstyle% \frac{1}{2},b-\textstyle\frac{1}{2}c-\textstyle\frac{1}{2},d-\textstyle\frac{1% }{2}){}p_{n+1}\!\left(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-% \textstyle\frac{1}{2},d-\textstyle\frac{1}{2}\right)}}} {\displaystyle \frac{\delta\left[\omega(x;a,b,c,d)\ctsHahn{n}@{x}{a}{b}{c}{d}\right]}{\delta x} {}=-(n+1)\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2} c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) {} \ctsHahn{n+1}@{x}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\textstyle\frac{1}{2}} }

Substitution(s): ω ⁢ ( x ; a , b , c , d ) = Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c - i ⁢ x ) ⁢ Γ ⁡ ( d - i ⁢ x ) 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d)=\Gamma\left(a+% \mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma\left(c-\mathrm{i}x% \right)\Gamma\left(d-\mathrm{i}x\right)}}}


Rodrigues-type formula

ω ⁢ ( x ; a , b , c , d ) ⁢ p n ⁡ ( x ; a , b , c , d ) = ( - 1 ) n n ! ⁢ ( δ δ ⁢ x ) n ⁢ [ ω ⁢ ( x ; a + 1 2 ⁢ n , b + 1 2 ⁢ n ⁢ c + 1 2 ⁢ n , d + 1 2 ⁢ n ) ] 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 superscript 1 𝑛 𝑛 superscript 𝛿 𝛿 𝑥 𝑛 delimited-[] 𝜔 𝑥 𝑎 1 2 𝑛 𝑏 1 2 𝑛 𝑐 1 2 𝑛 𝑑 1 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d)p_{n}\!\left(x;a,b,% c,d\right){}=\frac{(-1)^{n}}{n!}\left(\frac{\delta}{\delta x}\right)^{n}\left[% \omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}nc+\textstyle\frac{1}% {2}n,d+\textstyle\frac{1}{2}n)\right]}}} {\displaystyle \omega(x;a,b,c,d)\ctsHahn{n}@{x}{a}{b}{c}{d} {}=\frac{(-1)^n}{n!}\left(\frac{\delta}{\delta x}\right)^n \left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right] }

Substitution(s): ω ⁢ ( x ; a , b , c , d ) = Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c - i ⁢ x ) ⁢ Γ ⁡ ( d - i ⁢ x ) 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d)=\Gamma\left(a+% \mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma\left(c-\mathrm{i}x% \right)\Gamma\left(d-\mathrm{i}x\right)}}}


Generating functions

\HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ a + i ⁢ x ⁢ a + c - i ⁢ t ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ d - i ⁢ x ⁢ b + d ⁢ i ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c , d ) ( a + c ) n ⁢ ( b + d ) n ⁢ t n \HyperpFq 11 @ @ 𝑎 imaginary-unit 𝑥 𝑎 𝑐 imaginary-unit 𝑡 \HyperpFq 11 @ @ 𝑑 imaginary-unit 𝑥 𝑏 𝑑 imaginary-unit 𝑡 superscript subscript 𝑛 0 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑏 𝑑 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{1}{1}@@{a+\mathrm{i}x}{a+c% }{-\mathrm{i}t}\,\HyperpFq{1}{1}@@{d-\mathrm{i}x}{b+d}{\mathrm{i}t}=\sum_{n=0}% ^{\infty}\frac{p_{n}\!\left(x;a,b,c,d\right)}{{\left(a+c\right)_{n}}{\left(b+d% \right)_{n}}}t^{n}}}} {\displaystyle \HyperpFq{1}{1}@@{a+\iunit x}{a+c}{-\iunit t}\,\HyperpFq{1}{1}@@{d-\iunit x}{b+d}{\iunit t}= \sum_{n=0}^{\infty}\frac{\ctsHahn{n}@{x}{a}{b}{c}{d}}{\pochhammer{a+c}{n}\pochhammer{b+d}{n}}t^n }
\HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ a + i ⁢ x ⁢ a + d - i ⁢ t ⁢ \HyperpFq ⁢ 11 ⁢ @ ⁢ @ ⁢ c - i ⁢ x ⁢ b + c ⁢ i ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c , d ) ( a + d ) n ⁢ ( b + c ) n ⁢ t n \HyperpFq 11 @ @ 𝑎 imaginary-unit 𝑥 𝑎 𝑑 imaginary-unit 𝑡 \HyperpFq 11 @ @ 𝑐 imaginary-unit 𝑥 𝑏 𝑐 imaginary-unit 𝑡 superscript subscript 𝑛 0 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑑 𝑛 Pochhammer-symbol 𝑏 𝑐 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{1}{1}@@{a+\mathrm{i}x}{a+d% }{-\mathrm{i}t}\,\HyperpFq{1}{1}@@{c-\mathrm{i}x}{b+c}{\mathrm{i}t}=\sum_{n=0}% ^{\infty}\frac{p_{n}\!\left(x;a,b,c,d\right)}{{\left(a+d\right)_{n}}{\left(b+c% \right)_{n}}}t^{n}}}} {\displaystyle \HyperpFq{1}{1}@@{a+\iunit x}{a+d}{-\iunit t}\,\HyperpFq{1}{1}@@{c-\iunit x}{b+c}{\iunit t}= \sum_{n=0}^{\infty}\frac{\ctsHahn{n}@{x}{a}{b}{c}{d}}{\pochhammer{a+d}{n}\pochhammer{b+c}{n}}t^n }
( 1 - t ) 1 - a - b - c - d ⁢ \HyperpFq ⁢ 32 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ ( a + b + c + d - 1 ) , 1 2 ⁢ ( a + b + c + d ) , a + i ⁢ x ⁢ a + c , a + d - 4 ⁢ t ( 1 - t ) 2 = ∑ n = 0 ∞ ( a + b + c + d - 1 ) n ( a + c ) n ⁢ ( a + d ) n ⁢ i n ⁢ p n ⁡ ( x ; a , b , c , d ) ⁢ t n superscript 1 𝑡 1 𝑎 𝑏 𝑐 𝑑 \HyperpFq 32 @ @ 1 2 𝑎 𝑏 𝑐 𝑑 1 1 2 𝑎 𝑏 𝑐 𝑑 𝑎 imaginary-unit 𝑥 𝑎 𝑐 𝑎 𝑑 4 𝑡 superscript 1 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 1 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 imaginary-unit 𝑛 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-t)^{1-a-b-c-d}\,\HyperpFq{3}{2}@@% {\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+\mathrm{i}x}{a+c,a+d}{-\frac{4t% }{(1-t)^{2}}}{}=\sum_{n=0}^{\infty}\frac{{\left(a+b+c+d-1\right)_{n}}}{{\left(% a+c\right)_{n}}{\left(a+d\right)_{n}}{\mathrm{i}^{n}}}p_{n}\!\left(x;a,b,c,d% \right)t^{n}}}} {\displaystyle (1-t)^{1-a-b-c-d}\,\HyperpFq{3}{2}@@{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+\iunit x}{a+c,a+d}{-\frac{4t}{(1-t)^2}} {}=\sum_{n=0}^{\infty} \frac{\pochhammer{a+b+c+d-1}{n}}{\pochhammer{a+c}{n}\pochhammer{a+d}{n}\iunit^n}\ctsHahn{n}@{x}{a}{b}{c}{d}t^n }

Limit relations

Wilson polynomial to Continuous Hahn polynomial

lim t → ∞ ⁡ W n ⁡ ( ( x + t ) 2 ; a - i ⁢ t , b - i ⁢ t , c + i ⁢ t , d + i ⁢ t ) ( - 2 ⁢ t ) n ⁢ n ! = p n ⁡ ( x ; a , b , c , d ) subscript → 𝑡 Wilson-polynomial-W 𝑛 superscript 𝑥 𝑡 2 𝑎 imaginary-unit 𝑡 𝑏 imaginary-unit 𝑡 𝑐 imaginary-unit 𝑡 𝑑 imaginary-unit 𝑡 superscript 2 𝑡 𝑛 𝑛 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{W_{n}% \!\left((x+t)^{2};a-\mathrm{i}t,b-\mathrm{i}t,c+\mathrm{i}t,d+\mathrm{i}t% \right)}{(-2t)^{n}n!}=p_{n}\!\left(x;a,b,c,d\right)}}} {\displaystyle \lim_{t\rightarrow\infty} \frac{\Wilson{n}@{(x+t)^2}{a-\iunit t}{b-\iunit t}{c+\iunit t}{d+\iunit t}}{(-2t)^nn!}=\ctsHahn{n}@{x}{a}{b}{c}{d} }

Continuous Hahn polynomial to Meixner-Pollaczek polynomial

lim t → ∞ ⁡ p n ⁡ ( x + t ; λ - i ⁢ t , t ⁢ tan ⁡ ϕ , λ + i ⁢ t , t ⁢ tan ⁡ ϕ ) t n = P n ( λ ) ⁡ ( x ; ϕ ) ( cos ⁡ ϕ ) n subscript → 𝑡 continuous-Hahn-polynomial 𝑛 𝑥 𝑡 𝜆 imaginary-unit 𝑡 𝑡 italic-ϕ 𝜆 imaginary-unit 𝑡 𝑡 italic-ϕ superscript 𝑡 𝑛 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ superscript italic-ϕ 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{p_{n}% \!\left(x+t;\lambda-\mathrm{i}t,t\tan\phi,\lambda+\mathrm{i}t,t\tan\phi\right)% }{t^{n}}=\frac{P^{(\lambda)}_{n}\!\left(x;\phi\right)}{(\cos\phi)^{n}}}}} {\displaystyle \lim_{t\rightarrow\infty}\frac{\ctsHahn{n}@{x+t}{\lambda-\iunit t}{t\tan@@{\phi}}{\lambda+\iunit t}{t\tan@@{\phi}}}{t^n} =\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{(\cos@@{\phi})^n} }

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} }

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} }

Remark

( a + b ) n ⁢ ( a + c ) n ( a + b ) k ⁢ ( a + c ) k = ( a + b + k ) n - k ⁢ ( a + c + k ) n - k Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑏 𝑘 Pochhammer-symbol 𝑎 𝑐 𝑘 Pochhammer-symbol 𝑎 𝑏 𝑘 𝑛 𝑘 Pochhammer-symbol 𝑎 𝑐 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\frac{{\left(a+b\right)_{n}}{\left(a% +c\right)_{n}}}{{\left(a+b\right)_{k}}{\left(a+c\right)_{k}}}={\left(a+b+k% \right)_{n-k}}{\left(a+c+k\right)_{n-k}}}}} {\displaystyle \frac{\pochhammer{a+b}{n}\pochhammer{a+c}{n}}{\pochhammer{a+b}{k}\pochhammer{a+c}{k}}=\pochhammer{a+b+k}{n-k}\pochhammer{a+c+k}{n-k} }

Koornwinder Addendum: Continuous Hahn

Continuous Hahn: Special cases

p n ⁡ ( x ; a , a + 1 2 , a , a + 1 2 ) = ( 2 ⁢ a ) n ⁢ ( 2 ⁢ a + 1 2 ) n ( 4 ⁢ a ) n ⁢ P n ( 2 ⁢ a ) ⁡ ( 2 ⁢ x ; 1 2 ⁢ π ) continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑎 1 2 𝑎 𝑎 1 2 Pochhammer-symbol 2 𝑎 𝑛 Pochhammer-symbol 2 𝑎 1 2 𝑛 Pochhammer-symbol 4 𝑎 𝑛 Meixner-Pollaczek-polynomial-P 2 𝑎 𝑛 2 𝑥 1 2 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,a+\frac{1}{2},a,a+% \frac{1}{2}\right)=\frac{{\left(2a\right)_{n}}{\left(2a+\frac{1}{2}\right)_{n}% }}{{\left(4a\right)_{n}}}P^{(2a)}_{n}\!\left(2x;\frac{1}{2}\pi\right)}}} {\displaystyle \ctsHahn{n}@{x}{a}{a+\frac12}{a}{a+\frac12}= \frac{\pochhammer{2a}{n} \pochhammer{2a+\frac12}{n}}{\pochhammer{4a}{n}} \MeixnerPollaczek{2a}{n}@{2x}{\frac12\cpi} }

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