Continuous dual Hahn

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Continuous dual Hahn

Hypergeometric representation

S n ⁡ ( x 2 ; a , b , c ) ( a + b ) n ⁢ ( a + c ) n = \HyperpFq ⁢ 32 ⁢ @ ⁢ @ - n , a + i ⁢ x , a - i ⁢ x ⁢ a + b , a + c ⁢ 1 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 \HyperpFq 32 @ @ 𝑛 𝑎 imaginary-unit 𝑥 𝑎 imaginary-unit 𝑥 𝑎 𝑏 𝑎 𝑐 1 {\displaystyle{\displaystyle{\displaystyle\frac{S_{n}\!\left(x^{2};a,b,c\right% )}{{\left(a+b\right)_{n}}{\left(a+c\right)_{n}}}=\HyperpFq{3}{2}@@{-n,a+% \mathrm{i}x,a-\mathrm{i}x}{a+b,a+c}{1}}}} {\displaystyle \frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}}=\HyperpFq{3}{2}@@{-n,a+\iunit x,a-\iunit x}{a+b,a+c}{1} }

Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ 0 ∞ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 ⁢ S m ⁡ ( x 2 ; a , b , c ) ⁢ S n ⁡ ( x 2 ; a , b , c ) ⁢ 𝑑 x = Γ ⁡ ( n + a + b ) ⁢ Γ ⁡ ( n + a + c ) ⁢ Γ ⁡ ( n + b + c ) ⁢ n ! ⁢ δ m , n 1 2 superscript subscript 0 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 continuous-dual-Hahn-normalized-S 𝑚 superscript 𝑥 2 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 differential-d 𝑥 Euler-Gamma 𝑛 𝑎 𝑏 Euler-Gamma 𝑛 𝑎 𝑐 Euler-Gamma 𝑛 𝑏 𝑐 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}\left% |\frac{\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma% \left(c+\mathrm{i}x\right)}{\Gamma\left(2\mathrm{i}x\right)}\right|^{2}S_{m}\!% \left(x^{2};a,b,c\right)S_{n}\!\left(x^{2};a,b,c\right)\,dx{}=\Gamma\left(n+a+% b\right)\Gamma\left(n+a+c\right)\Gamma\left(n+b+c\right)n!\,\delta_{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_0^{\infty}\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2 \ctsdualHahn{m}@{x^2}{a}{b}{c}\ctsdualHahn{n}@{x^2}{a}{b}{c}\,dx {}=\EulerGamma@{n+a+b}\EulerGamma@{n+a+c}\EulerGamma@{n+b+c}n!\,\Kronecker{m}{n} }
1 2 ⁢ π ⁢ ∫ 0 ∞ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 ⁢ S m ⁡ ( x 2 ; a , b , c ) ⁢ S n ⁡ ( x 2 ; a , b , c ) ⁢ 𝑑 x + Γ ⁡ ( a + b ) ⁢ Γ ⁡ ( a + c ) ⁢ Γ ⁡ ( b - a ) ⁢ Γ ⁡ ( c - a ) Γ ⁡ ( - 2 ⁢ a ) ⁢ ∑ k = 0 , 1 , 2 ⁢ … a + k < 0 ( 2 ⁢ a ) k ⁢ ( a + 1 ) k ⁢ ( a + b ) k ⁢ ( a + c ) k ( a ) k ⁢ ( a - b + 1 ) k ⁢ ( a - c + 1 ) k ⁢ k ! ⁢ ( - 1 ) k ⁢ S m ⁡ ( - ( a + k ) 2 ; a , b , c ) ⁢ S n ⁡ ( - ( a + k ) 2 ; a , b , c ) = Γ ⁡ ( n + a + b ) ⁢ Γ ⁡ ( n + a + c ) ⁢ Γ ⁡ ( n + b + c ) ⁢ n ! ⁢ δ m , n 1 2 superscript subscript 0 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 continuous-dual-Hahn-normalized-S 𝑚 superscript 𝑥 2 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 differential-d 𝑥 Euler-Gamma 𝑎 𝑏 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑏 𝑎 Euler-Gamma 𝑐 𝑎 Euler-Gamma 2 𝑎 subscript 𝑘 0 1 2 … 𝑎 𝑘 0 Pochhammer-symbol 2 𝑎 𝑘 Pochhammer-symbol 𝑎 1 𝑘 Pochhammer-symbol 𝑎 𝑏 𝑘 Pochhammer-symbol 𝑎 𝑐 𝑘 Pochhammer-symbol 𝑎 𝑘 Pochhammer-symbol 𝑎 𝑏 1 𝑘 Pochhammer-symbol 𝑎 𝑐 1 𝑘 𝑘 superscript 1 𝑘 continuous-dual-Hahn-normalized-S 𝑚 superscript 𝑎 𝑘 2 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑎 𝑘 2 𝑎 𝑏 𝑐 Euler-Gamma 𝑛 𝑎 𝑏 Euler-Gamma 𝑛 𝑎 𝑐 Euler-Gamma 𝑛 𝑏 𝑐 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}\left% |\frac{\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma% \left(c+\mathrm{i}x\right)}{\Gamma\left(2\mathrm{i}x\right)}\right|^{2}S_{m}\!% \left(x^{2};a,b,c\right)S_{n}\!\left(x^{2};a,b,c\right)\,dx{}+\frac{\Gamma% \left(a+b\right)\Gamma\left(a+c\right)\Gamma\left(b-a\right)\Gamma\left(c-a% \right)}{\Gamma\left(-2a\right)}{}\sum_{\begin{array}[]{c}{\scriptstyle k=0,1,% 2\ldots}\\ {\scriptstyle a+k<0}\end{array}}\frac{{\left(2a\right)_{k}}{\left(a+1\right)_{% k}}{\left(a+b\right)_{k}}{\left(a+c\right)_{k}}}{{\left(a\right)_{k}}{\left(a-% b+1\right)_{k}}{\left(a-c+1\right)_{k}}k!}(-1)^{k}{}S_{m}\!\left(-(a+k)^{2};a,% b,c\right)S_{n}\!\left(-(a+k)^{2};a,b,c\right){}=\Gamma\left(n+a+b\right)% \Gamma\left(n+a+c\right)\Gamma\left(n+b+c\right)n!\,\delta_{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_0^{\infty}\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2 \ctsdualHahn{m}@{x^2}{a}{b}{c}\ctsdualHahn{n}@{x^2}{a}{b}{c}\,dx {}+\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{b-a}\EulerGamma@{c-a}}{\EulerGamma@{-2a}} {}\sum_{\begin{array}{c} {\scriptstyle k=0,1,2\ldots}\ {\scriptstyle a+k<0}\end{array}} \frac{\pochhammer{2a}{k}\pochhammer{a+1}{k}\pochhammer{a+b}{k}\pochhammer{a+c}{k}}{\pochhammer{a}{k}\pochhammer{a-b+1}{k}\pochhammer{a-c+1}{k}k!}(-1)^k {} \ctsdualHahn{m}@{-(a+k)^2}{a}{b}{c}\ctsdualHahn{n}@{-(a+k)^2}{a}{b}{c} {}=\EulerGamma@{n+a+b}\EulerGamma@{n+a+c}\EulerGamma@{n+b+c}n!\,\Kronecker{m}{n} }

Recurrence relation

- ( a 2 + x 2 ) ⁢ S ~ n ⁡ ( x 2 ) = A n ⁢ S ~ n + 1 ⁡ ( x 2 ) - ( A n + C n ) ⁢ S ~ n ⁡ ( x 2 ) + C n ⁢ S ~ n - 1 ⁡ ( x 2 ) superscript 𝑎 2 superscript 𝑥 2 continuous-dual-Hahn-normalized-S-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 subscript 𝐴 𝑛 continuous-dual-Hahn-normalized-S-tilde 𝑛 1 superscript 𝑥 2 𝑎 𝑏 𝑐 subscript 𝐴 𝑛 subscript 𝐶 𝑛 continuous-dual-Hahn-normalized-S-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 subscript 𝐶 𝑛 continuous-dual-Hahn-normalized-S-tilde 𝑛 1 superscript 𝑥 2 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle-\left(a^{2}+x^{2}\right){\tilde{S}}% _{n}\!\left(x^{2}\right)=A_{n}{\tilde{S}}_{n+1}\!\left(x^{2}\right)-\left(A_{n% }+C_{n}\right){\tilde{S}}_{n}\!\left(x^{2}\right)+C_{n}{\tilde{S}}_{n-1}\!% \left(x^{2}\right)}}} {\displaystyle -\left(a^2+x^2\right)\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}= A_n\normctsdualHahnStilde{n+1}@@{x^2}{a}{b}{c}-\left(A_n+C_n\right)\normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}+C_n\normctsdualHahnStilde{n-1}@@{x^2}{a}{b}{c} }

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


S ~ n ⁡ ( x 2 ) := S ~ n ⁡ ( x 2 ; a , b , c ) = S n ⁡ ( x 2 ; a , b , c ) ( a + b ) n ⁢ ( a + c ) n assign continuous-dual-Hahn-normalized-S-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{S}}_{n}\!\left(x^{2}\right):% ={\tilde{S}}_{n}\!\left(x^{2};a,b,c\right)=\frac{S_{n}\!\left(x^{2};a,b,c% \right)}{{\left(a+b\right)_{n}}{\left(a+c\right)_{n}}}}}} {\displaystyle \normctsdualHahnStilde{n}@@{x^2}{a}{b}{c}:=\normctsdualHahnStilde{n}@{x^2}{a}{b}{c}=\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}} }

Monic recurrence relation

x ⁢ S ^ n ⁡ ( x ) = S ^ n + 1 ⁡ ( x ) + ( A n + C n - a 2 ) ⁢ S ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ S ^ n - 1 ⁡ ( x ) 𝑥 continuous-dual-Hahn-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 continuous-dual-Hahn-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 subscript 𝐴 𝑛 subscript 𝐶 𝑛 superscript 𝑎 2 continuous-dual-Hahn-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 continuous-dual-Hahn-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle x{\widehat{S}}_{n}\!\left(x\right)=% {\widehat{S}}_{n+1}\!\left(x\right)+(A_{n}+C_{n}-a^{2}){\widehat{S}}_{n}\!% \left(x\right)+A_{n-1}C_{n}{\widehat{S}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicctsdualHahn{n}@@{x}{a}{b}{c}=\monicctsdualHahn{n+1}@@{x}{a}{b}{c}+(A_n+C_n-a^2)\monicctsdualHahn{n}@@{x}{a}{b}{c}+A_{n-1}C_n\monicctsdualHahn{n-1}@@{x}{a}{b}{c} }

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


S n ⁡ ( x 2 ; a , b , c ) = ( - 1 ) n ⁢ S ^ n ⁡ ( x 2 ) continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 superscript 1 𝑛 continuous-dual-Hahn-monic-p 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle S_{n}\!\left(x^{2};a,b,c\right)=(-1% )^{n}{\widehat{S}}_{n}\!\left(x^{2}\right)}}} {\displaystyle \ctsdualHahn{n}@{x^2}{a}{b}{c}=(-1)^n\monicctsdualHahn{n}@@{x^2}{a}{b}{c} }

Difference equation

n ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( x + i ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( x - i ) 𝑛 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 imaginary-unit delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 imaginary-unit {\displaystyle{\displaystyle{\displaystyle ny(x)=B(x)y(x+\mathrm{i})-\left[B(x% )+D(x)\right]y(x)+D(x)y(x-\mathrm{i})}}} {\displaystyle ny(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 ) ⁢ ( c + i ⁢ x ) 2 ⁢ i ⁢ x ⁢ ( 2 ⁢ i ⁢ x + 1 ) 𝐷 𝑥 𝑎 imaginary-unit 𝑥 𝑏 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 2 imaginary-unit 𝑥 2 imaginary-unit 𝑥 1 {\displaystyle{\displaystyle{\displaystyle D(x)=\frac{(a+\mathrm{i}x)(b+% \mathrm{i}x)(c+\mathrm{i}x)}{2\mathrm{i}x(2\mathrm{i}x+1)}}}} &

B ⁢ ( x ) = ( a - i ⁢ x ) ⁢ ( b - i ⁢ x ) ⁢ ( c - i ⁢ x ) 2 ⁢ i ⁢ x ⁢ ( 2 ⁢ i ⁢ x - 1 ) 𝐵 𝑥 𝑎 imaginary-unit 𝑥 𝑏 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 2 imaginary-unit 𝑥 2 imaginary-unit 𝑥 1 {\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(a-\mathrm{i}x)(b-% \mathrm{i}x)(c-\mathrm{i}x)}{2\mathrm{i}x(2\mathrm{i}x-1)}}}} &

y ⁢ ( x ) = S n ⁡ ( x 2 ; a , b , c ) 𝑦 𝑥 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle y(x)=S_{n}\!\left(x^{2};a,b,c\right% )}}}


Forward shift operator

S n ⁡ ( ( x + 1 2 ⁢ i ) 2 ; a , b , c ) - S n ⁡ ( ( x - 1 2 ⁢ i ) 2 ; a , b , c ) = - 2 ⁢ i ⁢ n ⁢ x ⁢ S n - 1 ⁡ ( x 2 ; a + 1 2 , b + 1 2 , c + 1 2 ) continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 2 imaginary-unit 𝑛 𝑥 continuous-dual-Hahn-normalized-S 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 {\displaystyle{\displaystyle{\displaystyle S_{n}\!\left((x+\textstyle\frac{1}{% 2}\mathrm{i})^{2};a,b,c\right)-S_{n}\!\left((x-\textstyle\frac{1}{2}\mathrm{i}% )^{2};a,b,c\right){}=-2\mathrm{i}nxS_{n-1}\!\left(x^{2};a+\textstyle\frac{1}{2% },b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2}\right)}}} {\displaystyle \ctsdualHahn{n}@{(x+\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c}-\ctsdualHahn{n}@{(x-\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c} {}=-2\iunit nx\ctsdualHahn{n-1}@{x^2}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}} }
δ ⁢ S n ⁡ ( x 2 ; a , b , c ) δ ⁢ x 2 = - n ⁢ S n - 1 ⁡ ( x 2 ; a + 1 2 , b + 1 2 , c + 1 2 ) 𝛿 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝛿 superscript 𝑥 2 𝑛 continuous-dual-Hahn-normalized-S 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 {\displaystyle{\displaystyle{\displaystyle\frac{\delta S_{n}\!\left(x^{2};a,b,% c\right)}{\delta x^{2}}=-nS_{n-1}\!\left(x^{2};a+\textstyle\frac{1}{2},b+% \textstyle\frac{1}{2},c+\textstyle\frac{1}{2}\right)}}} {\displaystyle \frac{\delta \ctsdualHahn{n}@{x^2}{a}{b}{c}}{\delta x^2}=-n\ctsdualHahn{n-1}@{x^2}{a+\textstyle\frac{1}{2}}{ b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}} }

Backward shift operator

( a - 1 2 - i ⁢ x ) ⁢ ( b - 1 2 - i ⁢ x ) ⁢ ( c - 1 2 - i ⁢ x ) ⁢ S n ⁡ ( ( x + 1 2 ⁢ i ) 2 ; a , b , c ) - ( a - 1 2 + i ⁢ x ) ⁢ ( b - 1 2 + i ⁢ x ) ⁢ ( c - 1 2 + i ⁢ x ) ⁢ S n ⁡ ( ( x - 1 2 ⁢ i ) 2 ; a , b , c ) = - 2 ⁢ i ⁢ x ⁢ S n + 1 ⁢ ( x 2 ; a - 1 2 , b - 1 2 ⁢ c - 1 2 ) 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 𝑐 1 2 imaginary-unit 𝑥 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 𝑐 1 2 imaginary-unit 𝑥 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 2 imaginary-unit 𝑥 subscript 𝑆 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 {\displaystyle{\displaystyle{\displaystyle(a-\textstyle\frac{1}{2}-\mathrm{i}x% )(b-\textstyle\frac{1}{2}-\mathrm{i}x)(c-\textstyle\frac{1}{2}-\mathrm{i}x)S_{% n}\!\left((x+\textstyle\frac{1}{2}\mathrm{i})^{2};a,b,c\right){}-(a-\textstyle% \frac{1}{2}+\mathrm{i}x)(b-\textstyle\frac{1}{2}+\mathrm{i}x)(c-\textstyle% \frac{1}{2}+\mathrm{i}x)S_{n}\!\left((x-\textstyle\frac{1}{2}\mathrm{i})^{2};a% ,b,c\right){}=-2\mathrm{i}xS_{n+1}(x^{2};a-\textstyle\frac{1}{2},b-\textstyle% \frac{1}{2}c-\textstyle\frac{1}{2})}}} {\displaystyle (a-\textstyle\frac{1}{2}-\iunit x)(b-\textstyle\frac{1}{2}-\iunit x) (c-\textstyle\frac{1}{2}-\iunit x) \ctsdualHahn{n}@{(x+\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c} {}-(a-\textstyle\frac{1}{2}+\iunit x)(b-\textstyle\frac{1}{2}+\iunit x) (c-\textstyle\frac{1}{2}+\iunit x) \ctsdualHahn{n}@{(x-\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c} {}=-2\iunit xS_{n+1}(x^2;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2} c-\textstyle\frac{1}{2}) }
δ ⁢ [ ω ⁢ ( x ; a , b , c ) ⁢ S n ⁡ ( x 2 ; a , b , c ) ] δ ⁢ x 2 = ω ⁢ ( x ; a - 1 2 , b - 1 2 , c - 1 2 ) ⁢ S n + 1 ⁡ ( x 2 ; a - 1 2 , b - 1 2 , c - 1 2 ) 𝛿 delimited-[] 𝜔 𝑥 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝛿 superscript 𝑥 2 𝜔 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 continuous-dual-Hahn-normalized-S 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 {\displaystyle{\displaystyle{\displaystyle\frac{\delta\left[\omega(x;a,b,c)S_{% n}\!\left(x^{2};a,b,c\right)\right]}{\delta x^{2}}{}=\omega(x;a-\textstyle% \frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2})S_{n+1}\!\left(x^{% 2};a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2}% \right)}}} {\displaystyle \frac{\delta\left[\omega(x;a,b,c)\ctsdualHahn{n}@{x^2}{a}{b}{c}\right]}{\delta x^2} {}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-\textstyle\frac{1}{2}) \ctsdualHahn{n+1}@{x^2}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}} }

Substitution(s): ω ⁢ ( x ; a , b , c ) = 1 2 ⁢ i ⁢ x ⁢ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 𝜔 𝑥 𝑎 𝑏 𝑐 1 2 imaginary-unit 𝑥 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c)=\frac{1}{2\mathrm{i}% x}\left|\frac{\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)% \Gamma\left(c+\mathrm{i}x\right)}{\Gamma\left(2\mathrm{i}x\right)}\right|^{2}}}}


Rodrigues-type formula

ω ⁢ ( x ; a , b , c ) ⁢ S n ⁡ ( x 2 ; a , b , c ) = ( δ δ ⁢ x 2 ) n ⁢ [ ω ⁢ ( x ; a + 1 2 ⁢ n ⁢ b + 1 2 ⁢ n , c + 1 2 ⁢ n ) ] 𝜔 𝑥 𝑎 𝑏 𝑐 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 superscript 𝛿 𝛿 superscript 𝑥 2 𝑛 delimited-[] 𝜔 𝑥 𝑎 1 2 𝑛 𝑏 1 2 𝑛 𝑐 1 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c)S_{n}\!\left(x^{2};a,% b,c\right)=\left(\frac{\delta}{\delta x^{2}}\right)^{n}\left[\omega(x;a+% \textstyle\frac{1}{2}nb+\textstyle\frac{1}{2}n,c+\textstyle\frac{1}{2}n)\right% ]}}} {\displaystyle \omega(x;a,b,c)\ctsdualHahn{n}@{x^2}{a}{b}{c}= \left(\frac{\delta}{\delta x^2}\right)^n\left[\omega(x;a+\textstyle\frac{1}{2}n b+\textstyle\frac{1}{2}n,c+\textstyle\frac{1}{2}n)\right] }

Substitution(s): ω ⁢ ( x ; a , b , c ) = 1 2 ⁢ i ⁢ x ⁢ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 𝜔 𝑥 𝑎 𝑏 𝑐 1 2 imaginary-unit 𝑥 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c)=\frac{1}{2\mathrm{i}% x}\left|\frac{\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)% \Gamma\left(c+\mathrm{i}x\right)}{\Gamma\left(2\mathrm{i}x\right)}\right|^{2}}}}


Generating functions

( 1 - t ) - c + i ⁢ x ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ a + i ⁢ x , b + i ⁢ x ⁢ a + b ⁢ t = ∑ n = 0 ∞ S n ⁡ ( x 2 ; a , b , c ) ( a + b ) n ⁢ n ! ⁢ t n superscript 1 𝑡 𝑐 imaginary-unit 𝑥 \HyperpFq 21 @ @ 𝑎 imaginary-unit 𝑥 𝑏 imaginary-unit 𝑥 𝑎 𝑏 𝑡 superscript subscript 𝑛 0 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑎 𝑏 𝑛 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-t)^{-c+\mathrm{i}x}\,\HyperpFq{2}% {1}@@{a+\mathrm{i}x,b+\mathrm{i}x}{a+b}{t}=\sum_{n=0}^{\infty}\frac{S_{n}\!% \left(x^{2};a,b,c\right)}{{\left(a+b\right)_{n}}n!}t^{n}}}} {\displaystyle (1-t)^{-c+\iunit x}\,\HyperpFq{2}{1}@@{a+\iunit x,b+\iunit x}{a+b}{t} =\sum_{n=0}^{\infty}\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}n!}t^n }
( 1 - t ) - b + i ⁢ x ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ a + i ⁢ x , c + i ⁢ x ⁢ a + c ⁢ t = ∑ n = 0 ∞ S n ⁡ ( x 2 ; a , b , c ) ( a + c ) n ⁢ n ! ⁢ t n superscript 1 𝑡 𝑏 imaginary-unit 𝑥 \HyperpFq 21 @ @ 𝑎 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 𝑎 𝑐 𝑡 superscript subscript 𝑛 0 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑎 𝑐 𝑛 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-t)^{-b+\mathrm{i}x}\,\HyperpFq{2}% {1}@@{a+\mathrm{i}x,c+\mathrm{i}x}{a+c}{t}=\sum_{n=0}^{\infty}\frac{S_{n}\!% \left(x^{2};a,b,c\right)}{{\left(a+c\right)_{n}}n!}t^{n}}}} {\displaystyle (1-t)^{-b+\iunit x}\,\HyperpFq{2}{1}@@{a+\iunit x,c+\iunit x}{a+c}{t} =\sum_{n=0}^{\infty}\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+c}{n}n!}t^n }
( 1 - t ) - a + i ⁢ x ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ b + i ⁢ x , c + i ⁢ x ⁢ b + c ⁢ t = ∑ n = 0 ∞ S n ⁡ ( x 2 ; a , b , c ) ( b + c ) n ⁢ n ! ⁢ t n superscript 1 𝑡 𝑎 imaginary-unit 𝑥 \HyperpFq 21 @ @ 𝑏 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 𝑏 𝑐 𝑡 superscript subscript 𝑛 0 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑏 𝑐 𝑛 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-t)^{-a+\mathrm{i}x}\,\HyperpFq{2}% {1}@@{b+\mathrm{i}x,c+\mathrm{i}x}{b+c}{t}=\sum_{n=0}^{\infty}\frac{S_{n}\!% \left(x^{2};a,b,c\right)}{{\left(b+c\right)_{n}}n!}t^{n}}}} {\displaystyle (1-t)^{-a+\iunit x}\,\HyperpFq{2}{1}@@{b+\iunit x,c+\iunit x}{b+c}{t} =\sum_{n=0}^{\infty}\frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{b+c}{n}n!}t^n }
e t ⁢ \HyperpFq ⁢ 22 ⁢ @ ⁢ @ ⁢ a + i ⁢ x , a - i ⁢ x ⁢ a + b , a + c - t = ∑ n = 0 ∞ S n ⁡ ( x 2 ; a , b , c ) ( a + b ) n ⁢ ( a + c ) n ⁢ n ! ⁢ t n 𝑡 \HyperpFq 22 @ @ 𝑎 imaginary-unit 𝑥 𝑎 imaginary-unit 𝑥 𝑎 𝑏 𝑎 𝑐 𝑡 superscript subscript 𝑛 0 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle{\mathrm{e}^{t}}\,\HyperpFq{2}{2}@@{% a+\mathrm{i}x,a-\mathrm{i}x}{a+b,a+c}{-t}=\sum_{n=0}^{\infty}\frac{S_{n}\!% \left(x^{2};a,b,c\right)}{{\left(a+b\right)_{n}}{\left(a+c\right)_{n}}n!}t^{n}% }}} {\displaystyle \expe^t\,\HyperpFq{2}{2}@@{a+\iunit x,a-\iunit x}{a+b,a+c}{-t}=\sum_{n=0}^{\infty} \frac{\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}n!}t^n }
( 1 - t ) - γ ⁢ \HyperpFq ⁢ 32 ⁢ @ ⁢ @ ⁢ γ , a + i ⁢ x , a - i ⁢ x ⁢ a + b , a + c ⁢ t t - 1 = ∑ n = 0 ∞ ( γ ) n ⁢ S n ⁡ ( x 2 ; a , b , c ) ( a + b ) n ⁢ ( a + c ) n ⁢ n ! ⁢ t n superscript 1 𝑡 𝛾 \HyperpFq 32 @ @ 𝛾 𝑎 imaginary-unit 𝑥 𝑎 imaginary-unit 𝑥 𝑎 𝑏 𝑎 𝑐 𝑡 𝑡 1 superscript subscript 𝑛 0 Pochhammer-symbol 𝛾 𝑛 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle(1-t)^{-\gamma}\,\HyperpFq{3}{2}@@{% \gamma,a+\mathrm{i}x,a-\mathrm{i}x}{a+b,a+c}{\frac{t}{t-1}}{}=\sum_{n=0}^{% \infty}\frac{{\left(\gamma\right)_{n}}S_{n}\!\left(x^{2};a,b,c\right)}{{\left(% a+b\right)_{n}}{\left(a+c\right)_{n}}n!}t^{n}}}} {\displaystyle (1-t)^{-\gamma}\,\HyperpFq{3}{2}@@{\gamma,a+\iunit x,a-\iunit x}{a+b,a+c}{\frac{t}{t-1}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{\gamma}{n}\ctsdualHahn{n}@{x^2}{a}{b}{c}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}n!}t^n }

Constraint(s): γ 𝛾 {\displaystyle{\displaystyle{\displaystyle\gamma}}} arbitrary


Limit relations

Wilson polynomial to Continuous dual Hahn polynomial

lim d → ∞ ⁡ W n ⁡ ( x 2 ; a , b , c , d ) ( a + d ) n = S n ⁡ ( x 2 ; a , b , c ) subscript → 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑑 𝑛 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle\lim_{d\rightarrow\infty}\frac{W_{n}% \!\left(x^{2};a,b,c,d\right)}{{\left(a+d\right)_{n}}}=S_{n}\!\left(x^{2};a,b,c% \right)}}} {\displaystyle \lim_{d\rightarrow\infty}\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+d}{n}}=\ctsdualHahn{n}@{x^2}{a}{b}{c} }

Continuous dual Hahn polynomial to Meixner-Pollaczek polynomial

lim t → ∞ ⁡ S n ⁡ ( ( x - t ) 2 ; λ + i ⁢ t , λ - i ⁢ t , t ⁢ cot ⁡ ϕ ) t n ⁢ n ! = P n ( λ ) ⁡ ( x ; ϕ ) ( sin ⁡ ϕ ) n subscript → 𝑡 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 𝑡 2 𝜆 imaginary-unit 𝑡 𝜆 imaginary-unit 𝑡 𝑡 italic-ϕ superscript 𝑡 𝑛 𝑛 Meixner-Pollaczek-polynomial-P 𝜆 𝑛 𝑥 italic-ϕ superscript italic-ϕ 𝑛 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{S_{n}% \!\left((x-t)^{2};\lambda+\mathrm{i}t,\lambda-\mathrm{i}t,t\cot\phi\right)}{t^% {n}n!}=\frac{P^{(\lambda)}_{n}\!\left(x;\phi\right)}{(\sin\phi)^{n}}}}} {\displaystyle \lim_{t\rightarrow\infty}\frac{\ctsdualHahn{n}@{(x-t)^2}{\lambda+\iunit t}{\lambda-\iunit t}{t\cot@@{\phi}}}{t^nn!} =\frac{\MeixnerPollaczek{\lambda}{n}@{x}{\phi}}{(\sin@@{\phi})^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 dual Hahn

Continuous dual Hahn: Special value

S n ⁡ ( - a 2 ; a , b , c ) = ( a + b ) n ⁢ ( a + c ) n continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑎 2 𝑎 𝑏 𝑐 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 {\displaystyle{\displaystyle{\displaystyle S_{n}\!\left(-a^{2};a,b,c\right)={% \left(a+b\right)_{n}}{\left(a+c\right)_{n}}}}} {\displaystyle \ctsdualHahn{n}@{-a^2}{a}{b}{c}=\pochhammer{a+b}{n}\pochhammer{a+c}{n} }

Uniqueness of orthogonality measure

| S n ⁡ ( - a 2 ; a , b , c ) | 2 h n = O ⁢ ( n 2 ⁢ ℜ ⁡ a - 1 ) ⁢ as  n → ∞ . superscript continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑎 2 𝑎 𝑏 𝑐 2 subscript ℎ 𝑛 𝑂 superscript 𝑛 2 𝑎 1 as  n → ∞ . {\displaystyle{\displaystyle{\displaystyle\frac{|S_{n}\!\left(-a^{2};a,b,c% \right)|^{2}}{h_{n}}=O(n^{2\Re{a}-1})\hbox{as $n\to\infty$.}}}} {\displaystyle \frac{|\ctsdualHahn{n}@{-a^2}{a}{b}{c}|^2}{h_n} = O(n^{2\realpart{a}-1}) \hbox{as $n\to\infty$.} }

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