Continuous dual q-Hahn

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Continuous dual q-Hahn

Basic hypergeometric representation

a n ⁢ p n ⁡ ( x ; a , b , c | q ) ( a ⁢ b , a ⁢ c ; q ) n = \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ ⁢ a ⁢ b , a ⁢ c ⁢ q ⁢ q superscript 𝑎 𝑛 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑞 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 imaginary-unit 𝜃 𝑎 imaginary-unit 𝜃 𝑎 𝑏 𝑎 𝑐 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle\frac{a^{n}p_{n}\!\left(x;a,b,c|q% \right)}{\left(ab,ac;q\right)_{n}}=\qHyperrphis{3}{2}@@{q^{-n},a{\mathrm{e}^{% \mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab,ac}{q}{q}}}} {\displaystyle \frac{a^n\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,ac}{q}{n}} =\qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{ab,ac}{q}{q} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Orthogonality relation(s)

1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ p m ⁡ ( x ; a , b , c | q ) ⁢ p n ⁡ ( x ; a , b , c | q ) ⁢ 𝑑 x = h n ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-dual-q-Hahn-polynomial-p 𝑚 𝑥 𝑎 𝑏 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 differential-d 𝑥 subscript ℎ 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}p_{m}\!\left(x;a,b,c|q\right)p_{n}\!\left(x;a,b,c|q\right)\,% dx=h_{n}\,\delta_{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsdualqHahn{m}@{x}{a}{b}{c}{q}\ctsdualqHahn{n}@{x}{a}{b}{c}{q}\,dx =h_n\,\Kronecker{m}{n} }

Substitution(s): h n = 1 ( q n + 1 , a ⁢ b ⁢ q n , a ⁢ c ⁢ q n , b ⁢ c ⁢ q n ; q ) ∞ subscript ℎ 𝑛 1 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 𝑎 𝑐 superscript 𝑞 𝑛 𝑏 𝑐 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{1}{\left(q^{n+1},abq^{n% },acq^{n},bcq^{n};q\right)_{\infty}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; a , b , c | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}% \theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})% h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


1 2 ⁢ π ⁢ ∫ - 1 1 w ⁢ ( x ) 1 - x 2 ⁢ p m ⁡ ( x ; a , b , c | q ) ⁢ p n ⁡ ( x ; a , b , c | q ) ⁢ 𝑑 x + ∑ k 1 < a ⁢ q k ≤ a w k ⁢ p m ⁡ ( x k ; a , b , c | q ) ⁢ p n ⁡ ( x k ; a , b , c | q ) = h n ⁢ δ m , n 1 2 superscript subscript 1 1 𝑤 𝑥 1 superscript 𝑥 2 continuous-dual-q-Hahn-polynomial-p 𝑚 𝑥 𝑎 𝑏 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 differential-d 𝑥 subscript 𝑘 1 𝑎 superscript 𝑞 𝑘 𝑎 subscript 𝑤 𝑘 continuous-dual-q-Hahn-polynomial-p 𝑚 subscript 𝑥 𝑘 𝑎 𝑏 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 subscript 𝑥 𝑘 𝑎 𝑏 𝑐 𝑞 subscript ℎ 𝑛 Kronecker-delta 𝑚 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x% )}{\sqrt{1-x^{2}}}p_{m}\!\left(x;a,b,c|q\right)p_{n}\!\left(x;a,b,c|q\right)\,% dx{}+\sum_{\begin{array}[]{c}\scriptstyle k\\ \scriptstyle 1<aq^{k}\leq a\end{array}}w_{k}p_{m}\!\left(x_{k};a,b,c|q\right)p% _{n}\!\left(x_{k};a,b,c|q\right)=h_{n}\,\delta_{m,n}}}} {\displaystyle \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\ctsdualqHahn{m}@{x}{a}{b}{c}{q}\ctsdualqHahn{n}@{x}{a}{b}{c}{q}\,dx {}+\sum_{\begin{array}{c}\scriptstyle k\ \scriptstyle 1

Substitution(s): w k = ( a - 2 ; q ) ∞ ( q , a ⁢ b , a ⁢ c , a - 1 ⁢ b , a - 1 ⁢ c ; q ) ∞ ⁢ ( 1 - a 2 ⁢ q 2 ⁢ k ) ⁢ ( a 2 , a ⁢ b , a ⁢ c ; q ) k ( 1 - a 2 ) ⁢ ( q , a ⁢ b - 1 ⁢ q , a ⁢ c - 1 ⁢ q ; q ) k ⁢ ( - 1 ) k ⁢ q - \binomial ⁢ k ⁢ 2 ⁢ ( 1 a 2 ⁢ b ⁢ c ) k subscript 𝑤 𝑘 q-Pochhammer-symbol superscript 𝑎 2 𝑞 q-Pochhammer-symbol 𝑞 𝑎 𝑏 𝑎 𝑐 superscript 𝑎 1 𝑏 superscript 𝑎 1 𝑐 𝑞 1 superscript 𝑎 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol superscript 𝑎 2 𝑎 𝑏 𝑎 𝑐 𝑞 𝑘 1 superscript 𝑎 2 q-Pochhammer-symbol 𝑞 𝑎 superscript 𝑏 1 𝑞 𝑎 superscript 𝑐 1 𝑞 𝑞 𝑘 superscript 1 𝑘 superscript 𝑞 \binomial 𝑘 2 superscript 1 superscript 𝑎 2 𝑏 𝑐 𝑘 {\displaystyle{\displaystyle{\displaystyle w_{k}=\frac{\left(a^{-2};q\right)_{% \infty}}{\left(q,ab,ac,a^{-1}b,a^{-1}c;q\right)_{\infty}}\frac{(1-a^{2}q^{2k})% \left(a^{2},ab,ac;q\right)_{k}}{(1-a^{2})\left(q,ab^{-1}q,ac^{-1}q;q\right)_{k% }}(-1)^{k}q^{-\binomial{k}{2}}\left(\frac{1}{a^{2}bc}\right)^{k}}}} &

x k = a ⁢ q k + ( a ⁢ q k ) - 1 2 subscript 𝑥 𝑘 𝑎 superscript 𝑞 𝑘 superscript 𝑎 superscript 𝑞 𝑘 1 2 {\displaystyle{\displaystyle{\displaystyle x_{k}=\frac{aq^{k}+\left(aq^{k}% \right)^{-1}}{2}}}} &
h n = 1 ( q n + 1 , a ⁢ b ⁢ q n , a ⁢ c ⁢ q n , b ⁢ c ⁢ q n ; q ) ∞ subscript ℎ 𝑛 1 q-Pochhammer-symbol superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 𝑎 𝑐 superscript 𝑞 𝑛 𝑏 𝑐 superscript 𝑞 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle h_{n}=\frac{1}{\left(q^{n+1},abq^{n% },acq^{n},bcq^{n};q\right)_{\infty}}}}} &
w ⁢ ( x ) := w ⁢ ( x ; a , b , c | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}% \theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})% h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Recurrence relation

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

Substitution(s): C n = a ⁢ ( 1 - q n ) ⁢ ( 1 - b ⁢ c ⁢ q n - 1 ) subscript 𝐶 𝑛 𝑎 1 superscript 𝑞 𝑛 1 𝑏 𝑐 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=a(1-q^{n})(1-bcq^{n-1})}}} &
A n = a - 1 ⁢ ( 1 - a ⁢ b ⁢ q n ) ⁢ ( 1 - a ⁢ c ⁢ q n ) subscript 𝐴 𝑛 superscript 𝑎 1 1 𝑎 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑐 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle A_{n}=a^{-1}(1-abq^{n})(1-acq^{n})}}}


p ~ n ⁡ ( x ) := p ~ n ⁡ ( x ; a , b , c | q ) = a n ⁢ p n ⁡ ( x ; a , b , c | q ) ( a ⁢ b , a ⁢ c ; q ) n assign continuous-dual-q-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-normalized-p-tilde 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 superscript 𝑎 𝑛 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{p}}_{n}\!\left(x\right):={% \tilde{p}}_{n}\!\left(x;a,b,c|q\right)=\frac{a^{n}p_{n}\!\left(x;a,b,c|q\right% )}{\left(ab,ac;q\right)_{n}}}}} {\displaystyle \normctsdualqHahnptilde{n}@@{x}{a}{b}{c}{q}:=\normctsdualqHahnptilde{n}@{x}{a}{b}{c}{q}=\frac{a^n\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,ac}{q}{n}} }

Monic recurrence relation

x ⁢ p ^ n ⁡ ( x ) = p ^ n + 1 ⁡ ( x ) + 1 2 ⁢ [ a + a - 1 - ( A n + C n ) ] ⁢ p ^ n ⁡ ( x ) + 1 4 ⁢ ( 1 - q n ) ⁢ ( 1 - a ⁢ b ⁢ q n - 1 ) ⁢ ( 1 - a ⁢ c ⁢ q n - 1 ) ⁢ ( 1 - b ⁢ c ⁢ q n - 1 ) ⁢ p ^ n - 1 ⁡ ( x ) 𝑥 continuous-dual-q-Hahn-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑞 1 2 delimited-[] 𝑎 superscript 𝑎 1 subscript 𝐴 𝑛 subscript 𝐶 𝑛 continuous-dual-q-Hahn-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 1 4 1 superscript 𝑞 𝑛 1 𝑎 𝑏 superscript 𝑞 𝑛 1 1 𝑎 𝑐 superscript 𝑞 𝑛 1 1 𝑏 𝑐 superscript 𝑞 𝑛 1 continuous-dual-q-Hahn-polynomial-monic-p 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle x{\widehat{p}}_{n}\!\left(x\right)=% {\widehat{p}}_{n+1}\!\left(x\right)+\frac{1}{2}\left[a+a^{-1}-(A_{n}+C_{n})% \right]{\widehat{p}}_{n}\!\left(x\right){}+\frac{1}{4}(1-q^{n})(1-abq^{n-1}){}% (1-acq^{n-1})(1-bcq^{n-1}){\widehat{p}}_{n-1}\!\left(x\right)}}} {\displaystyle x\monicctsdualqHahn{n}@@{x}{a}{b}{c}{q}=\monicctsdualqHahn{n+1}@@{x}{a}{b}{c}{q}+\frac{1}{2}\left[a+a^{-1}-(A_n+C_n)\right]\monicctsdualqHahn{n}@@{x}{a}{b}{c}{q} {}+\frac{1}{4}(1-q^n)(1-abq^{n-1}) {}(1-acq^{n-1})(1-bcq^{n-1})\monicctsdualqHahn{n-1}@@{x}{a}{b}{c}{q} }

Substitution(s): C n = a ⁢ ( 1 - q n ) ⁢ ( 1 - b ⁢ c ⁢ q n - 1 ) subscript 𝐶 𝑛 𝑎 1 superscript 𝑞 𝑛 1 𝑏 𝑐 superscript 𝑞 𝑛 1 {\displaystyle{\displaystyle{\displaystyle C_{n}=a(1-q^{n})(1-bcq^{n-1})}}} &
A n = a - 1 ⁢ ( 1 - a ⁢ b ⁢ q n ) ⁢ ( 1 - a ⁢ c ⁢ q n ) subscript 𝐴 𝑛 superscript 𝑎 1 1 𝑎 𝑏 superscript 𝑞 𝑛 1 𝑎 𝑐 superscript 𝑞 𝑛 {\displaystyle{\displaystyle{\displaystyle A_{n}=a^{-1}(1-abq^{n})(1-acq^{n})}}}


p n ⁡ ( x ; a , b , c | q ) = 2 n ⁢ p ^ n ⁡ ( x ) continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 superscript 2 𝑛 continuous-dual-q-Hahn-polynomial-monic-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c|q\right)=2^{n}% {\widehat{p}}_{n}\!\left(x\right)}}} {\displaystyle \ctsdualqHahn{n}@{x}{a}{b}{c}{q}=2^n\monicctsdualqHahn{n}@@{x}{a}{b}{c}{q} }

q-Difference equation

( 1 - q ) 2 ⁢ D q ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 ⁢ c ⁢ q 1 2 | q ) ⁢ D q ⁢ y ⁢ ( x ) ] + 4 ⁢ q - n + 1 ⁢ ( 1 - q n ) ⁢ w ~ ⁢ ( x ; a , b , c | q ) ⁢ y ⁢ ( x ) = 0 superscript 1 𝑞 2 subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 conditional 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑞 subscript 𝐷 𝑞 𝑦 𝑥 4 superscript 𝑞 𝑛 1 1 superscript 𝑞 𝑛 ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 𝑦 𝑥 0 {\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;aq% ^{\frac{1}{2}},bq^{\frac{1}{2}}cq^{\frac{1}{2}}|q)D_{q}y(x)\right]{}+4q^{-n+1}% (1-q^{n}){\tilde{w}}(x;a,b,c|q)y(x)=0}}} {\displaystyle (1-q)^2D_q\left[{\tilde w}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}} cq^{\frac{1}{2}}|q)D_qy(x)\right] {}+4q^{-n+1}(1-q^n){\tilde w}(x;a,b,c|q)y(x)=0 }

Substitution(s): w ~ ⁢ ( x ; a , b , c | q ) := w ⁢ ( x ; a , b , c | q ) 1 - x 2 assign ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c|q):=\frac{w(x;a,% b,c|q)}{\sqrt{1-x^{2}}}}}} &

y ⁢ ( x ) = p n ⁡ ( x ; a , b , c | q ) 𝑦 𝑥 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle y(x)=p_{n}\!\left(x;a,b,c|q\right)}}} &
w ⁢ ( x ) := w ⁢ ( x ; a , b , c | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}% \theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})% h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


q - n ⁢ ( 1 - q n ) ⁢ 𝒫 n ⁢ ( z ) = A ⁢ ( z ) ⁢ 𝒫 n ⁢ ( q ⁢ z ) - [ A ⁢ ( z ) + A ⁢ ( z - 1 ) ] ⁢ 𝒫 n ⁢ ( z ) + A ⁢ ( z - 1 ) ⁢ 𝒫 n ⁢ ( q - 1 ⁢ z ) superscript 𝑞 𝑛 1 superscript 𝑞 𝑛 subscript 𝒫 𝑛 𝑧 𝐴 𝑧 subscript 𝒫 𝑛 𝑞 𝑧 delimited-[] 𝐴 𝑧 𝐴 superscript 𝑧 1 subscript 𝒫 𝑛 𝑧 𝐴 superscript 𝑧 1 subscript 𝒫 𝑛 superscript 𝑞 1 𝑧 {\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n}){\mathcal{P}}_{n}(z)% =A(z){\mathcal{P}}_{n}(qz)-\left[A(z)+A(z^{-1})\right]{\mathcal{P}}_{n}(z)+A(z% ^{-1}){\mathcal{P}}_{n}(q^{-1}z)}}} {\displaystyle q^{-n}(1-q^n){\mathcal P}_n(z)=A(z){\mathcal P}_n(qz)-\left[A(z)+A(z^{-1})\right]{\mathcal P}_n(z) +A(z^{-1}){\mathcal P}_n(q^{-1}z) }

Substitution(s): A ⁢ ( z ) = ( 1 - a ⁢ z ) ⁢ ( 1 - b ⁢ z ) ⁢ ( 1 - c ⁢ z ) ( 1 - z 2 ) ⁢ ( 1 - q ⁢ z 2 ) 𝐴 𝑧 1 𝑎 𝑧 1 𝑏 𝑧 1 𝑐 𝑧 1 superscript 𝑧 2 1 𝑞 superscript 𝑧 2 {\displaystyle{\displaystyle{\displaystyle A(z)=\frac{(1-az)(1-bz)(1-cz)}{(1-z% ^{2})(1-qz^{2})}}}} &
𝒫 n ⁢ ( z ) := ( a ⁢ b , a ⁢ c ; q ) n a n ⁢ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ q - n , a ⁢ z , a ⁢ z - 1 ⁢ a ⁢ b , a ⁢ c ⁢ q ⁢ q assign subscript 𝒫 𝑛 𝑧 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑞 𝑛 superscript 𝑎 𝑛 \qHyperrphis 32 @ @ superscript 𝑞 𝑛 𝑎 𝑧 𝑎 superscript 𝑧 1 𝑎 𝑏 𝑎 𝑐 𝑞 𝑞 {\displaystyle{\displaystyle{\displaystyle{\mathcal{P}}_{n}(z):=\frac{\left(ab% ,ac;q\right)_{n}}{a^{n}}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{ab,ac}{q}{q}% }}}


Forward shift operator

δ q ⁢ p n ⁡ ( x ; a , b , c | q ) = - q - 1 2 ⁢ n ⁢ ( 1 - q n ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ p n - 1 ⁡ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 , c ⁢ q 1 2 | q ) subscript 𝛿 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 superscript 𝑞 1 2 𝑛 1 superscript 𝑞 𝑛 imaginary-unit 𝜃 imaginary-unit 𝜃 continuous-dual-q-Hahn-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}p_{n}\!\left(x;a,b,c|q% \right)=-q^{-\frac{1}{2}n}(1-q^{n})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e% }^{-\mathrm{i}\theta}}){}p_{n-1}\!\left(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}},cq% ^{\frac{1}{2}}|q\right)}}} {\displaystyle \delta_q \ctsdualqHahn{n}@{x}{a}{b}{c}{q}=-q^{-\frac{1}{2}n}(1-q^n)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {} \ctsdualqHahn{n-1}@{x}{aq^{\frac{1}{2}}}{bq^{\frac{1}{2}}}{cq^{\frac{1}{2}}}{q} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


D q ⁢ p n ⁡ ( x ; a , b , c | q ) = 2 ⁢ q - 1 2 ⁢ ( n - 1 ) ⁢ 1 - q n 1 - q ⁢ p n - 1 ⁡ ( x ; a ⁢ q 1 2 , b ⁢ q 1 2 , c ⁢ q 1 2 | q ) subscript 𝐷 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 2 superscript 𝑞 1 2 𝑛 1 1 superscript 𝑞 𝑛 1 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}p_{n}\!\left(x;a,b,c|q\right)=% 2q^{-\frac{1}{2}(n-1)}\frac{1-q^{n}}{1-q}p_{n-1}\!\left(x;aq^{\frac{1}{2}},bq^% {\frac{1}{2}},cq^{\frac{1}{2}}|q\right)}}} {\displaystyle D_q \ctsdualqHahn{n}@{x}{a}{b}{c}{q}= 2q^{-\frac{1}{2}(n-1)}\frac{1-q^n}{1-q}\ctsdualqHahn{n-1}@{x}{aq^{\frac{1}{2}}}{bq^{\frac{1}{2}}}{cq^{\frac{1}{2}}}{q} }

Backward shift operator

δ q ⁢ [ w ~ ⁢ ( x ; a , b , c | q ) ⁢ p n ⁡ ( x ; a , b , c | q ) ] = q - 1 2 ⁢ ( n + 1 ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 , c ⁢ q - 1 2 | q ) ⁢ p n + 1 ⁡ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 , c ⁢ q - 1 2 | q ) subscript 𝛿 𝑞 delimited-[] ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 superscript 𝑞 1 2 𝑛 1 imaginary-unit 𝜃 imaginary-unit 𝜃 ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 conditional 𝑐 superscript 𝑞 1 2 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;a,b,c|% q)p_{n}\!\left(x;a,b,c|q\right)\right]{}=q^{-\frac{1}{2}(n+1)}({\mathrm{e}^{% \mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}}){\tilde{w}}(x;aq^{-\frac{1% }{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}}|q){}p_{n+1}\!\left(x;aq^{-\frac{1}{2% }},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}}|q\right)}}} {\displaystyle \delta_q\left[{\tilde w}(x;a,b,c|q)\ctsdualqHahn{n}@{x}{a}{b}{c}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}}|q) {} \ctsdualqHahn{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{q} }

Substitution(s): w ~ ⁢ ( x ; a , b , c | q ) := w ⁢ ( x ; a , b , c | q ) 1 - x 2 assign ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c|q):=\frac{w(x;a,% b,c|q)}{\sqrt{1-x^{2}}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; a , b , c | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}% \theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})% h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


D q ⁢ [ w ~ ⁢ ( x ; a , b , c | q ) ⁢ p n ⁡ ( x ; a , b , c | q ) ] = - 2 ⁢ q - 1 2 ⁢ n 1 - q ⁢ w ~ ⁢ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 , c ⁢ q - 1 2 | q ) ⁢ p n + 1 ⁡ ( x ; a ⁢ q - 1 2 , b ⁢ q - 1 2 , c ⁢ q - 1 2 | q ) subscript 𝐷 𝑞 delimited-[] ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 2 superscript 𝑞 1 2 𝑛 1 𝑞 ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 conditional 𝑐 superscript 𝑞 1 2 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 1 𝑥 𝑎 superscript 𝑞 1 2 𝑏 superscript 𝑞 1 2 𝑐 superscript 𝑞 1 2 𝑞 {\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;a,b,c|q)p_% {n}\!\left(x;a,b,c|q\right)\right]{}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde{w}% }(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}}|q)p_{n+1}\!\left(x;% aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}}|q\right)}}} {\displaystyle D_q\left[{\tilde w}(x;a,b,c|q)\ctsdualqHahn{n}@{x}{a}{b}{c}{q}\right] {}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}}|q) \ctsdualqHahn{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{q} }

Substitution(s): w ~ ⁢ ( x ; a , b , c | q ) := w ⁢ ( x ; a , b , c | q ) 1 - x 2 assign ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c|q):=\frac{w(x;a,% b,c|q)}{\sqrt{1-x^{2}}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; a , b , c | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}% \theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})% h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Rodrigues-type formula

w ~ ⁢ ( x ; a , b , c | q ) ⁢ p n ⁡ ( x ; a , b , c | q ) = ( q - 1 2 ) n ⁢ q 1 4 ⁢ n ⁢ ( n - 1 ) ⁢ ( D q ) n ⁢ [ w ~ ⁢ ( x ; a ⁢ q 1 2 ⁢ n , b ⁢ q 1 2 ⁢ n , c ⁢ q 1 2 ⁢ n | q ) ] ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 superscript 𝑞 1 2 𝑛 superscript 𝑞 1 4 𝑛 𝑛 1 superscript subscript 𝐷 𝑞 𝑛 delimited-[] ~ 𝑤 𝑥 𝑎 superscript 𝑞 1 2 𝑛 𝑏 superscript 𝑞 1 2 𝑛 conditional 𝑐 superscript 𝑞 1 2 𝑛 𝑞 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c|q)p_{n}\!\left(x% ;a,b,c|q\right){}=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\left(D_{% q}\right)^{n}\left[{\tilde{w}}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n},cq^{\frac% {1}{2}n}|q)\right]}}} {\displaystyle {\tilde w}(x;a,b,c|q)\ctsdualqHahn{n}@{x}{a}{b}{c}{q} {}=\left(\frac{q-1}{2}\right)^nq^{\frac{1}{4}n(n-1)}\left(D_q\right)^n\left[{\tilde w}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n},cq^{\frac{1}{2}n}|q)\right] }

Substitution(s): w ~ ⁢ ( x ; a , b , c | q ) := w ⁢ ( x ; a , b , c | q ) 1 - x 2 assign ~ 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 1 superscript 𝑥 2 {\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b,c|q):=\frac{w(x;a,% b,c|q)}{\sqrt{1-x^{2}}}}}} &

w ⁢ ( x ) := w ⁢ ( x ; a , b , c | q ) = | ( e 2 ⁢ i ⁢ θ ; q ) ∞ ( a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ; q ) ∞ | 2 = h ⁢ ( x , 1 ) ⁢ h ⁢ ( x , - 1 ) ⁢ h ⁢ ( x , q 1 2 ) ⁢ h ⁢ ( x , - q 1 2 ) h ⁢ ( x , a ) ⁢ h ⁢ ( x , b ) ⁢ h ⁢ ( x , c ) assign 𝑤 𝑥 𝑤 𝑥 𝑎 𝑏 conditional 𝑐 𝑞 superscript q-Pochhammer-symbol 2 imaginary-unit 𝜃 𝑞 q-Pochhammer-symbol 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑞 2 ℎ 𝑥 1 ℎ 𝑥 1 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 superscript 𝑞 1 2 ℎ 𝑥 𝑎 ℎ 𝑥 𝑏 ℎ 𝑥 𝑐 {\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{% \left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{% \mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}% \theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})% h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)h(x,c)}}}} &
h ⁢ ( x , α ) := ∏ k = 0 ∞ ( 1 - 2 ⁢ α ⁢ x ⁢ q k + α 2 ⁢ q 2 ⁢ k ) = ( α ⁢ e i ⁢ θ , α ⁢ e - i ⁢ θ ; q ) ∞ assign ℎ 𝑥 𝛼 superscript subscript product 𝑘 0 1 2 𝛼 𝑥 superscript 𝑞 𝑘 superscript 𝛼 2 superscript 𝑞 2 𝑘 q-Pochhammer-symbol 𝛼 imaginary-unit 𝜃 𝛼 imaginary-unit 𝜃 𝑞 {\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}% \left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{% \mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}} &

x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Generating functions

( c ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ , b ⁢ e i ⁢ θ ⁢ a ⁢ b ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c | q ) ( a ⁢ b , q ; q ) n ⁢ t n q-Pochhammer-symbol 𝑐 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ 𝑎 imaginary-unit 𝜃 𝑏 imaginary-unit 𝜃 𝑎 𝑏 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 q-Pochhammer-symbol 𝑎 𝑏 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(ct;q\right)_{\infty}}{% \left({\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}\ \qHyperrphis{2}{1}@% @{a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}}{ab}{q}{{% \mathrm{e}^{-\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{p_{n}\!\left(x;a,% b,c|q\right)}{\left(ab,q;q\right)_{n}}t^{n}}}} {\displaystyle \frac{\qPochhammer{ct}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},b\expe^{\iunit\theta}}{ab}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ab,q}{q}{n}}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


( b ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ⁢ a ⁢ c ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c | q ) ( a ⁢ c , q ; q ) n ⁢ t n q-Pochhammer-symbol 𝑏 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ 𝑎 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑎 𝑐 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 q-Pochhammer-symbol 𝑎 𝑐 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(bt;q\right)_{\infty}}{% \left({\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}\ \qHyperrphis{2}{1}@% @{a{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}\theta}}}{ac}{q}{{% \mathrm{e}^{-\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{p_{n}\!\left(x;a,% b,c|q\right)}{\left(ac,q;q\right)_{n}}t^{n}}}} {\displaystyle \frac{\qPochhammer{bt}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},c\expe^{\iunit\theta}}{ac}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{ac,q}{q}{n}}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


( a ⁢ t ; q ) ∞ ( e i ⁢ θ ⁢ t ; q ) ∞ ⁢ \qHyperrphis ⁢ 21 ⁢ @ ⁢ @ ⁢ b ⁢ e i ⁢ θ , c ⁢ e i ⁢ θ ⁢ b ⁢ c ⁢ q ⁢ e - i ⁢ θ ⁢ t = ∑ n = 0 ∞ p n ⁡ ( x ; a , b , c | q ) ( b ⁢ c , q ; q ) n ⁢ t n q-Pochhammer-symbol 𝑎 𝑡 𝑞 q-Pochhammer-symbol imaginary-unit 𝜃 𝑡 𝑞 \qHyperrphis 21 @ @ 𝑏 imaginary-unit 𝜃 𝑐 imaginary-unit 𝜃 𝑏 𝑐 𝑞 imaginary-unit 𝜃 𝑡 superscript subscript 𝑛 0 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 q-Pochhammer-symbol 𝑏 𝑐 𝑞 𝑞 𝑛 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\frac{\left(at;q\right)_{\infty}}{% \left({\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}\ \qHyperrphis{2}{1}@% @{b{\mathrm{e}^{\mathrm{i}\theta}},c{\mathrm{e}^{\mathrm{i}\theta}}}{bc}{q}{{% \mathrm{e}^{-\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{p_{n}\!\left(x;a,% b,c|q\right)}{\left(bc,q;q\right)_{n}}t^{n}}}} {\displaystyle \frac{\qPochhammer{at}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{b\expe^{\iunit\theta},c\expe^{\iunit\theta}}{bc}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}{\qPochhammer{bc,q}{q}{n}}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


( t ; q ) ∞ ⋅ \qHyperrphis ⁢ 32 ⁢ @ ⁢ @ ⁢ a ⁢ e i ⁢ θ , a ⁢ e - i ⁢ θ , 0 ⁢ a ⁢ b , a ⁢ c ⁢ q ⁢ t = ∑ n = 0 ∞ ( - 1 ) n ⁢ a n ⁢ q \binomial ⁢ n ⁢ 2 ( a ⁢ b , a ⁢ c , q ; q ) n ⁢ p n ⁡ ( x ; a , b , c | q ) ⁢ t n ⋅ q-Pochhammer-symbol 𝑡 𝑞 \qHyperrphis 32 @ @ 𝑎 imaginary-unit 𝜃 𝑎 imaginary-unit 𝜃 0 𝑎 𝑏 𝑎 𝑐 𝑞 𝑡 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑎 𝑛 superscript 𝑞 \binomial 𝑛 2 q-Pochhammer-symbol 𝑎 𝑏 𝑎 𝑐 𝑞 𝑞 𝑛 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 superscript 𝑡 𝑛 {\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot% \qHyperrphis{3}{2}@@{a{\mathrm{e}^{\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i% }\theta}},0}{ab,ac}{q}{t}{}=\sum_{n=0}^{\infty}\frac{(-1)^{n}a^{n}q^{\binomial% {n}{2}}}{\left(ab,ac,q;q\right)_{n}}p_{n}\!\left(x;a,b,c|q\right)t^{n}}}} {\displaystyle \label{GenContDualqHahn4} \qPochhammer{t}{q}{\infty}\cdot\qHyperrphis{3}{2}@@{a\expe^{\iunit\theta},a\expe^{-\iunit\theta},0}{ab,ac}{q}{t} {}=\sum_{n=0}^{\infty}\frac{(-1)^na^nq^{\binomial{n}{2}}}{\qPochhammer{ab,ac,q}{q}{n}} \ctsdualqHahn{n}@{x}{a}{b}{c}{q}t^n }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


Limit relations

Askey-Wilson polynomial to Continuous dual q-Hahn polynomial

p n ⁡ ( x ; a , b , c , 0 | q ) = p n ⁡ ( x ; a , b , c | q ) Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 0 𝑞 continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c,0\,|\,q\right)% =p_{n}\!\left(x;a,b,c|q\right)}}} {\displaystyle \AskeyWilson{n}@{x}{a}{b}{c}{0}{q}=\ctsdualqHahn{n}@{x}{a}{b}{c}{q} }

Continuous dual q-Hahn polynomial to Al-Salam-Chihara polynomial

p n ⁡ ( x ; a , b , 0 | q ) = Q n ⁡ ( x ; a , b | q ) continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 0 𝑞 Al-Salam-Chihara-polynomial-Q 𝑛 𝑥 𝑎 𝑏 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,0|q\right)=Q_{n}% \!\left(x;a,b\,|\,q\right)}}} {\displaystyle \ctsdualqHahn{n}@{x}{a}{b}{0}{q}=\AlSalamChihara{n}@{x}{a}{b}{q} }

Continuous dual q-Hahn polynomial to Continuous dual Hahn polynomial

lim q → 1 ⁡ p n ⁡ ( 1 2 ⁢ ( q i ⁢ x + q - i ⁢ x ) ; q a , q b , q c | q ) ( 1 - q ) 2 ⁢ n = S n ⁡ ( x 2 ; a , b , c ) subscript → 𝑞 1 continuous-dual-q-Hahn-polynomial-p 𝑛 1 2 superscript 𝑞 imaginary-unit 𝑥 superscript 𝑞 imaginary-unit 𝑥 superscript 𝑞 𝑎 superscript 𝑞 𝑏 superscript 𝑞 𝑐 𝑞 superscript 1 𝑞 2 𝑛 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 {\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{p_{n}\!% \left(\frac{1}{2}\left(q^{\mathrm{i}x}+q^{-\mathrm{i}x}\right);q^{a},q^{b},q^{% c}|q\right)}{(1-q)^{2n}}=S_{n}\!\left(x^{2};a,b,c\right)}}} {\displaystyle \lim_{q\rightarrow 1}\frac{\ctsdualqHahn{n}@{\frac{1}{2}\left(q^{\iunit x}+q^{-\iunit x}\right)}{q^a}{q^b}{q^c}{q}}{(1-q)^{2n}}=\ctsdualHahn{n}@{x^2}{a}{b}{c} }

Koornwinder Addendum: Continuous dual q-Hahn

p n ⁡ ( x ; a , b , c | q ) := p n ⁡ ( x ; a , b , c , 0 | q ) assign continuous-dual-q-Hahn-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 𝑞 Askey-Wilson-polynomial-p 𝑛 𝑥 𝑎 𝑏 𝑐 0 𝑞 {\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,c|q\right):=p_{n% }\!\left(x;a,b,c,0\,|\,q\right)}}} {\displaystyle \ctsdualqHahn{n}@{x}{a}{b}{c }{q}:=\AskeyWilson{n}@{x}{a}{b}{c}{0 }{ q} }

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