Continuous dual q-Hahn

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Continuous dual q-Hahn

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Orthogonality relation(s)

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_n=\frac{1}{\qPochhammer{q^{n+1},abq^n,acq^n,bcq^n}{q}{\infty}}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta},c\expe^{\iunit\theta}}{q}{\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)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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<aq^k\leq a\end{array}} w_k\ctsdualqHahn{m}@{x_k}{a}{b}{c}{q}\ctsdualqHahn{n}@{x_k}{a}{b}{c}{q}=h_n\,\Kronecker{m}{n} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w_k=\frac{\qPochhammer{a^{-2}}{q}{\infty}}{\qPochhammer{q,ab,ac,a^{-1}b,a^{-1}c}{q}{\infty}} \frac{(1-a^2q^{2k})\qPochhammer{a^2,ab,ac}{q}{k}}{(1-a^2)\qPochhammer{q,ab^{-1}q,ac^{-1}q}{q}{k}} (-1)^kq^{-\binomial{k}{2}}\left(\frac{1}{a^2bc}\right)^k}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x_k=\frac{aq^k+\left(aq^k\right)^{-1}}{2}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h_n=\frac{1}{\qPochhammer{q^{n+1},abq^n,acq^n,bcq^n}{q}{\infty}}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta},c\expe^{\iunit\theta}}{q}{\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)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=a(1-q^n)(1-bcq^{n-1})}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=a^{-1}(1-abq^n)(1-acq^n)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=a(1-q^n)(1-bcq^{n-1})}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=a^{-1}(1-abq^n)(1-acq^n)}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsdualqHahn{n}@{x}{a}{b}{c}{q}=2^n\monicctsdualqHahn{n}@@{x}{a}{b}{c}{q} }}

q-Difference equation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;a,b,c|q):=\frac{w(x;a,b,c|q)}{\sqrt{1-x^2}}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\ctsdualqHahn{n}@{x}{a}{b}{c}{q}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta},c\expe^{\iunit\theta}}{q}{\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)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A(z)=\frac{(1-az)(1-bz)(1-cz)}{(1-z^2)(1-qz^2)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\mathcal P}_n(z):=\frac{\qPochhammer{ab,ac}{q}{n}}{a^n}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{ab,ac}{q}{q}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;a,b,c|q):=\frac{w(x;a,b,c|q)}{\sqrt{1-x^2}}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta},c\expe^{\iunit\theta}}{q}{\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)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;a,b,c|q):=\frac{w(x;a,b,c|q)}{\sqrt{1-x^2}}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta},c\expe^{\iunit\theta}}{q}{\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)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Rodrigues-type formula

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;a,b,c|q):=\frac{w(x;a,b,c|q)}{\sqrt{1-x^2}}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;a,b,c|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta},c\expe^{\iunit\theta}}{q}{\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)}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right) =\qPochhammer{\alpha\expe^{\iunit\theta},\alpha\expe^{-\iunit\theta}}{q}{\infty}}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Generating functions

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x=\cos@@{\theta}}}


Limit relations

Askey-Wilson polynomial to Continuous dual q-Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsdualqHahn{n}@{x}{a}{b}{0}{q}=\AlSalamChihara{n}@{x}{a}{b}{q} }}

Continuous dual q-Hahn polynomial to Continuous dual Hahn polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\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

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsdualqHahn{n}@{x}{a}{b}{c }{q}:=\AskeyWilson{n}@{x}{a}{b}{c}{0 }{ q} }}

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