Continuous q-Jacobi: Special cases
Continuous q-Jacobi: Special cases
Continuous q-ultraspherical / Rogers
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}}\beta^{-\frac{1}{2}n}\,\qHyperrphis{4}{3}@@{q^{-n},\beta^2q^n \beta^{\frac{1}{2}}\expe^{\iunit\theta},\beta^{\frac{1}{2}}\expe^{-\iunit\theta}}{\beta q^{\frac{1}{2}} -\beta,-\beta q^{\frac{1}{2}}}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}}\beta^{-n}\expe^{-\iunit n\theta}\,\qHyperrphis{3}{2}@@{q^{-n},\beta,\beta\expe^{2\iunit\theta}}{\beta^2,0}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\frac{\qPochhammer{\beta}{q}{n}}{\qPochhammer{q}{q}{n}}\expe^{\iunit n\theta}\,\qHyperrphis{2}{1}@@{q^{-n},\beta}{\beta^{-1}q^{-n+1}}{q}{\beta^{-1}q\expe^{-2\iunit\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}}\ctsqUltra{m}@{x}{\beta}{q}\ctsqUltra{n}@{x}{\beta}{q}\,dx {}=\frac{\qPochhammer{\beta,\beta q}{q}{\infty}}{\qPochhammer{\beta^2,q}{q}{\infty}} \frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}}\frac{(1-\beta)}{(1-\beta q^n)}\,\Kronecker{m}{n} }}
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}}}
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Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2(1-\beta q^n)x\ctsqUltra{n}@{x}{\beta}{q}=(1-q^{n+1})\ctsqUltra{n+1}@{x}{\beta}{q} {}+(1-\beta^2 q^{n-1})\ctsqUltra{n-1}@{x}{\beta}{q} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicctsqUltra{n}@@{x}{\beta}{q}=\monicctsqUltra{n+1}@@{x}{\beta}{q}+ \frac{(1-q^n)(1-\beta^2q^{n-1})}{4(1-\beta q^{n-1})(1-\beta q^n)}\monicctsqUltra{n-1}@@{x}{\beta}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\frac{2^n\qPochhammer{\beta}{q}{n}}{\qPochhammer{q}{q}{n}}\monicctsqUltra{n}@@{x}{\beta}{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;\beta q|q)D_qy(x)\right]+ \lambda_n{\tilde w}(x;\beta|q)y(x)=0 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;\beta|q):=\frac{w(x;\beta|q)}{\sqrt{1-x^2}}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\ctsqUltra{n}@{x}{\beta}{q}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;\beta|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{\beta^{\frac{1}{2}}\expe^{\iunit\theta},\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta} -\beta^{\frac{1}{2}}\expe^{\iunit\theta},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{\beta\expe^{2\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,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}}) h(x,-\beta^{\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}}
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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}}}
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Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_q\ctsqUltra{n}@{x}{\beta}{q}=-q^{-\frac{1}{2}n}(1-\beta)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) \ctsqUltra{n-1}@{x}{\beta q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D_q\ctsqUltra{n}@{x}{\beta}{q}=2q^{-\frac{1}{2}(n-1)}\frac{1-\beta}{1-q}\ctsqUltra{n-1}@{x}{\beta q}{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;\beta|q)\ctsqUltra{n}@{x}{\beta}{q}\right] {}=q^{-\frac{1}{2}(n+1)}\frac{(1-q^{n+1})(1-\beta^2q^{n-1})}{(1-\beta q^{-1})} (\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {}{\tilde w}(x;\beta q^{-1}|q)\ctsqUltra{n+1}@{x}{\beta q^{-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;\beta|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{\beta^{\frac{1}{2}}\expe^{\iunit\theta},\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta} -\beta^{\frac{1}{2}}\expe^{\iunit\theta},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{\beta\expe^{2\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,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}}) h(x,-\beta^{\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}}
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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}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D_q\left[{\tilde w}(x;\beta|q)\ctsqUltra{n}@{x}{\beta}{q}\right] {}=-\frac{2q^{-\frac{1}{2}n}(1-q^{n+1})(1-\beta^2q^{n-1})}{(1-q)(1-\beta q^{-1})} {}{\tilde w}(x;\beta q^{-1}|q)\ctsqUltra{n+1}@{x}{\beta q^{-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;\beta|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{\beta^{\frac{1}{2}}\expe^{\iunit\theta},\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta} -\beta^{\frac{1}{2}}\expe^{\iunit\theta},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{\beta\expe^{2\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,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}}) h(x,-\beta^{\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}}
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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}}}
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Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;\beta|q)\ctsqUltra{n}@{x}{\beta}{q} {}=\left(\frac{q-1}{2}\right)^n q^{\frac{1}{4}n(n-1)}\frac{\qPochhammer{\beta}{q}{n}}{\qPochhammer{q,\beta^2q^n}{q}{n}} \left(D_q\right)^n\left[{\tilde w}(x;\beta q^n|q)\right] }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;\beta|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{\beta^{\frac{1}{2}}\expe^{\iunit\theta},\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta} -\beta^{\frac{1}{2}}\expe^{\iunit\theta},-\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{\beta\expe^{2\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,\beta^{\frac{1}{2}})h(x,\beta^{\frac{1}{2}}q^{\frac{1}{2}}) h(x,-\beta^{\frac{1}{2}})h(x,-\beta^{\frac{1}{2}}q^{\frac{1}{2}})}}}
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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}}}
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Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left|\frac{\qPochhammer{\beta\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\right|^2 =\frac{\qPochhammer{\beta\expe^{\iunit\theta}t,\beta\expe^{-\iunit\theta}t}{q}{\infty}} {\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\ctsqUltra{n}@{x}{\beta}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{\beta,\beta\expe^{2\iunit\theta}}{\beta^2}{q}{\expe^{-\iunit\theta}t} =\sum_{n=0}^{\infty}\frac{\ctsqUltra{n}@{x}{\beta}{q}}{\qPochhammer{\beta^2}{q}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{\expe^{-\iunit\theta}t}{q}{\infty}\cdot \qHyperrphis{2}{1}@@{\beta,\beta\expe^{2\iunit\theta}}{\beta^2}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{(-1)^n\beta^nq^{\binomial{n}{2}}}{\qPochhammer{\beta^2}{q}{n}}\ctsqUltra{n}@{x}{\beta}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}\expe^{\iunit\theta} \beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta}}{\beta q^{\frac{1}{2}}}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{-\beta^{\frac{1}{2}}\expe^{-\iunit\theta} -\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{-\iunit\theta}}{\beta q^{\frac{1}{2}}}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-\beta,-\beta q^{\frac{1}{2}}}{q}{n}} {\qPochhammer{\beta^2,\beta q^{\frac{1}{2}}}{q}{n}}\ctsqUltra{n}@{x}{\beta}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}\expe^{\iunit\theta} -\beta^{\frac{1}{2}}\expe^{\iunit\theta}}{-\beta}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{-\iunit\theta} -\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{-\iunit\theta}}{-\beta q}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\beta q^{\frac{1}{2}},-\beta q^{\frac{1}{2}}}{q}{n}} {\qPochhammer{\beta^2,-\beta q}{q}{n}}\ctsqUltra{n}@{x}{\beta}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}\expe^{\iunit\theta} -\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{\iunit\theta}}{-\beta q^{\frac{1}{2}}}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{\beta^{\frac{1}{2}}q^{\frac{1}{2}}\expe^{-\iunit\theta} -\beta^{\frac{1}{2}}\expe^{-\iunit\theta}}{-\beta q^{\frac{1}{2}}}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-\beta,\beta q^{\frac{1}{2}}}{q}{n}} {\qPochhammer{\beta^2,-\beta q^{\frac{1}{2}}}{q}{n}}\ctsqUltra{n}@{x}{\beta}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{\gamma\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,\beta,\beta\expe^{2\iunit\theta}}{\beta^2,\gamma\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{\beta^2}{q}{n}}\ctsqUltra{n}@{x}{\beta}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary
Limit relations
Askey-Wilson polynomial to Continuous q-ultraspherical / Rogers polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{\beta^2}{q}{n}\AskeyWilson{n}@{x}{\beta^{\frac{1}{2}}}{\beta^{\frac{1}{2}}q^{\frac{1}{2}}}{ -\beta^{\frac{1}{2}}}{-\beta^{\frac{1}{2}}q^{\frac{1}{2}}}{q}} {\qPochhammer{\beta q^{\frac{1}{2}},-\beta,-\beta q^{\frac{1}{2}},q}{q}{n}}=\ctsqUltra{n}@{x}{\beta}{q} }}
q-Meixner-Pollaczek polynomial to Continuous q-ultraspherical /
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qMeixnerPollaczek{n}@{\cos@@{\phi}}{\beta}{q}=\ctsqUltra{n}@{\cos@@{\phi}}{\beta}{q} }}
Continuous q-ultraspherical / Rogers polynomial to Continuous q-Hermite polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\beta\rightarrow 0}\ctsqUltra{n}@{x}{\beta}{q}=\frac{\ctsqHermite{n}@{x}{q}}{\qPochhammer{q}{q}{n}} }}
Continuous q-ultraspherical / Rogers polynomial to Gegenbauer /
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\ctsqUltra{n}@{x}{q^{\lambda}}{q}=\Ultra{\lambda}{n}@{x} }}
Remarks
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{\beta}{q}=\sum_{k=0}^n\frac{\qPochhammer{\beta}{q}{k}\qPochhammer{\beta}{q}{n-k}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}} \expe^{\iunit(n-2k)\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqJacobi{\alpha}{\alpha}{n}@{x}{q}\stackrel{q^{\alpha+\frac{1}{2}}\rightarrow\beta}{\longrightarrow} \frac{\qPochhammer{\beta q^{\frac{1}{2}}}{q}{n}}{\qPochhammer{\beta^2}{q}{n}}\beta^{\frac{1}{2}n}\ctsqUltra{n}@{x}{\beta}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{q^{\alpha+\frac{1}{2}}}{q}= \frac{\qPochhammer{q^{2\alpha+1}}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}\ctsqJacobi{\alpha}{\alpha}{n}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{\beta}{q^{-1}}=(\beta q)^n\ctsqUltra{n}@{x}{\beta^{-1}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{q}{q}=\frac{\sin@{n+1}\theta}{\sin@@{\theta}}=\ChebyU{n}@{x} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{\beta\rightarrow 1}\frac{1-q^n}{2(1-\beta)}\ctsqUltra{n}@{x}{\beta}{q}=\cos@@{n}\theta =\ChebyT{n}@{x},\quad x=\cos@@{\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{2n}@{x}{q^{\lambda}}{q}=\frac{\qPochhammer{q^{\lambda},-q}{q}{n}} {\qPochhammer{q^{\frac{1}{2}},-q^{\frac{1}{2}}}{q}{n}}q^{-\frac{1}{2}n} \ctsqJacobiRahman{\lambda-\frac{1}{2}}{-\frac{1}{2}}{n}@{2x^2-1}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{2n+1}@{x}{q^{\lambda}}{q}=\frac{\qPochhammer{q^{\lambda},-1}{q}{n+1}} {\qPochhammer{q^{\frac{1}{2}},-q^{\frac{1}{2}}}{q}{n+1}}q^{-\frac{1}{2}n} x\ctsqJacobiRahman{\lambda-\frac{1}{2}}{\frac{1}{2}}{n}@{2x^2-1}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{x}{q^{\frac{1}{2}}}{q}=q^{-\frac{1}{4}n}\ctsqLegendre{n}@{x}{q} }}
Koornwinder Addendum: Continuous q-ultraspherical / Rogers
Re: (14.10.17)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{\cos@@{\theta}}{\beta }{ q}= \frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}} \beta^{-\frac12 n} \qHyperrphis{4}{3}@@{q^{-\frac12 n},\beta q^{\frac12 n},\beta^\frac12 \expe^{\iunit\theta},\beta^\frac12 \expe^{-\iunit\theta}}{-\beta,\beta^\frac12 q^{\frac14},-\beta^\frac12 q^{\frac14}}{q^\frac12}{q^\frac12} }}
Continuous q-Jacobi: Special cases: Special value
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{\frac12(\beta^\frac12+\beta^{-\frac12})}{\beta }{ q} =\frac{\qPochhammer{\beta^2}{q}{n}}{\qPochhammer{q}{q}{n}} \beta^{-\frac12 n} }}
Re: (14.10.21)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1-\beta z^2}{1-z^2} \ctsqUltrae{n}@{q^\frac12 z}{\beta }{ q}+ \frac{1-\beta z^{-2}}{1-z^{-2}} \ctsqUltrae{n}@{q^{-\frac12}z}{\beta }{ q}= (q^{-\frac12 n}+q^{\frac12 n} \beta) \ctsqUltrae{n}@{z}{\beta }{ q} }}
Re: (14.10.23)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltrae{n}@{q^\frac12 z}{\beta }{ q}-\ctsqUltrae{n}@{q^{-\frac12}z}{\beta }{ q}= q^{-\frac12 n}(\beta-1)(z-z^{-1})\ctsqUltrae{n-1}@{z}{q\beta }{ q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (\beta+1)\ctsqUltrae{n}@{q^\frac12 z}{\beta }{ q}=(q^{-\frac12 n}+q^{\frac12 n}\beta)\ctsqUltrae{n}@{z}{\beta }{ q} +q^{-\frac12 n}(\beta-1)(z-\beta z^{-1})\ctsqUltrae{n-1}@{z}{q\beta }{ q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle (\beta+1)\ctsqUltrae{n}@{q^{-\frac12}z}{\beta }{ q}=(q^{-\frac12 n}+q^{\frac12 n}\beta)\ctsqUltrae{n}@{z}{\beta }{ q} +q^{-\frac12 n}(\beta-1)(z^{-1}-\beta z)\ctsqUltrae{n-1}@{z}{q\beta }{ q} }}
Trigonometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqUltra{n}@{\cos@@{\theta}}{\beta }{ q}=\sum_{k=0}^n \frac{\qPochhammer{\beta}{q}{k} \qPochhammer{\beta}{q}{n-k}}{\qPochhammer{q}{q}{k} \qPochhammer{q}{q}{n-k}} \expe^{\iunit(n-2k)\theta} }}
Limit for q\downarrow-1
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\uparrow1} \ctsqUltra{2m}@{x}{-q^\la }{ -q}= \Ultra{\half(\la+1)}{m}@{2x^2-1}+\Ultra{\half(\la+1)}{m-1}@{2x^2-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\uparrow1} \ctsqUltra{2m+1}@{x}{-q^\la }{ -q}= 2x \Ultra{\half(\la+1)}{m}@{2x^2-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\uparrow1} \ctsqUltra{2m}@{x}{-q^\lambda }{ -q}= \frac{\pochhammer{\lambda}{m}}{\pochhammer{\frac12\lambda}{m}} \Jacobi{\frac12\lambda}{\frac12\lambda-1}{m}@{2x^2-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\uparrow1} \ctsqUltra{2m+1}@{x}{-q^\lambda }{ -q}= 2 \frac{\pochhammer{\lambda+1}{m}}{\pochhammer{\frac12\lambda+1}{m}} x \Jacobi{\frac12\lambda}{\frac12\lambda}{m}@{2x^2-1} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\uparrow1} \ctsqUltra{n}@{x}{-q^\lambda }{ -q} =\mathrm const \GenGegenbauer{\frac12\lambda}{\frac12\lambda-1}{n}@{x} }}
Continuous q-Legendre
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqLegendre{n}@{x}{q}=\qHyperrphis{4}{3}@@{q^{-n},q^{n+1},q^{\frac{1}{4}}\expe^{\iunit\theta} q^{\frac{1}{4}}\expe^{-\iunit\theta}}{q,-q^{\frac{1}{2}},-q}{q}{q} }}
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;1|q)}{\sqrt{1-x^2}}\ctsqLegendre{m}@{x}{q}\ctsqLegendre{n}@{x}{q}\,dx {}=\frac{\qPochhammer{q^{\frac{1}{2}}}{q}{\infty}}{\qPochhammer{q,q,-q^{\frac{1}{2}},-q}{q}{\infty}} \frac{q^{\frac{1}{2}n}}{1-q^{n+\frac{1}{2}}}\,\Kronecker{m}{n} }}
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}}}
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Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2(1-q^{n+\frac{1}{2}})x\ctsqLegendre{n}@{x}{q}=q^{-\frac{1}{4}}(1-q^{n+1})\ctsqLegendre{n+1}@{x}{q} +q^{\frac{1}{4}}(1-q^n)\ctsqLegendre{n-1}@{x}{q} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicctsqLegendre{n}@@{x}{q}=\monicctsqLegendre{n+1}@@{x}{q}+\frac{(1-q^n)^2}{4(1-q^{n-\frac{1}{2}})(1-q^{n+\frac{1}{2}})}\monicctsqLegendre{n-1}@@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqLegendre{n}@{x}{q}=\frac{2^nq^{\frac{1}{4}n}\qPochhammer{q^{\frac{1}{2}}}{q}{n}}{\qPochhammer{q}{q}{n}}\monicctsqLegendre{n}@@{x}{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;q^{\frac{1}{2}}|q)D_qy(x)\right]+ \lambda_n{\tilde w}(x;1|q)y(x)=0 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;a|q):=\frac{w(x;a|q)}{\sqrt{1-x^2}}}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lambda_n=4q^{-n+1}(1-q^n)(1-q^{n+1})}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{aq^{\frac{1}{4}}\expe^{\iunit\theta} aq^{\frac{3}{4}}\expe^{\iunit\theta},-aq^{\frac{1}{4}}\expe^{\iunit\theta} -aq^{\frac{3}{4}}\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{\iunit\theta},-\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}} {\qPochhammer{aq^{\frac{1}{4}}\expe^{\iunit\theta},-aq^{\frac{1}{4}}\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a^2q^{\frac{1}{2}}\expe^{2\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,aq^{\frac{1}{4}})h(x,aq^{\frac{3}{4}})h(x,-aq^{\frac{1}{4}})h(x,-aq^{\frac{3}{4}})}}}
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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}}}
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Rodrigues-type formula
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;1|q)\ctsqLegendre{n}@{x}{q}=\left(\frac{q-1}{2}\right)^n \frac{q^{\frac{1}{4}n^2}}{\qPochhammer{q,-q^{\frac{1}{2}},-q}{q}{n}} \left(D_q\right)^n\left[{\tilde w}(x;q^{\frac{1}{2}n}|q)\right] }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x;a|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{aq^{\frac{1}{4}}\expe^{\iunit\theta} aq^{\frac{3}{4}}\expe^{\iunit\theta},-aq^{\frac{1}{4}}\expe^{\iunit\theta} -aq^{\frac{3}{4}}\expe^{\iunit\theta}}{q}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{\iunit\theta},-\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}} {\qPochhammer{aq^{\frac{1}{4}}\expe^{\iunit\theta},-aq^{\frac{1}{4}}\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}\right|^2 =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a^2q^{\frac{1}{2}}\expe^{2\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,aq^{\frac{1}{4}})h(x,aq^{\frac{3}{4}})h(x,-aq^{\frac{1}{4}})h(x,-aq^{\frac{3}{4}})}}}
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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}}}
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Generating functions
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \left|\frac{\qPochhammer{q^{\frac{1}{2}}\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\right|^2 =\frac{\qPochhammer{q^{\frac{1}{2}}\expe^{\iunit\theta}t,q^{\frac{1}{2}}\expe^{-\iunit\theta}t}{q}{\infty}} {\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\frac{\ctsqLegendre{n}@{x}{q}}{q^{\frac{1}{4}n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{2}{1}@@{q^{\frac{1}{2}},q^{\frac{1}{2}}\expe^{2\iunit\theta}}{q}{q}{\expe^{-\iunit\theta}t} =\sum_{n=0}^{\infty}\frac{\ctsqLegendre{n}@{x}{q}}{\qPochhammer{q}{q}{n}q^{\frac{1}{4}n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{\expe^{-\iunit\theta}t}{q}{\infty}\cdot \qHyperrphis{2}{1}@@{q^{\frac{1}{2}},q^{\frac{1}{2}}\expe^{2\iunit\theta}}{q}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\frac{1}{4}n+\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}\ctsqLegendre{n}@{x}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{q^{\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{3}{4}}\expe^{\iunit\theta}}{q}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{-q^{\frac{1}{4}}\expe^{-\iunit\theta},-q^{\frac{3}{4}}\expe^{-\iunit\theta}}{q}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-q^{\frac{1}{2}},-q}{q}{n}}{\qPochhammer{q,q}{q}{n}} \frac{\ctsqLegendre{n}@{x}{q}}{q^{\frac{1}{4}n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{q^{\frac{1}{4}}\expe^{\iunit\theta},-q^{\frac{1}{4}}\expe^{\iunit\theta}}{-q^{\frac{1}{2}}}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{q^{\frac{3}{4}}\expe^{-\iunit\theta},-q^{\frac{3}{4}}\expe^{-\iunit\theta}}{-q^{\frac{3}{2}}}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-q}{q}{n}}{\qPochhammer{-q^{\frac{3}{2}}}{q}{n}} \frac{\ctsqLegendre{n}@{x}{q}}{q^{\frac{1}{4}n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{q^{\frac{1}{4}}\expe^{\iunit\theta},-q^{\frac{3}{4}}\expe^{\iunit\theta}}{-q}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{q^{\frac{3}{4}}\expe^{-\iunit\theta},-q^{\frac{1}{4}}\expe^{-\iunit\theta}}{-q}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-q^{\frac{1}{2}}}{q}{n}}{\qPochhammer{-q}{q}{n}} \frac{\ctsqLegendre{n}@{x}{q}}{q^{\frac{1}{4}n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\qPochhammer{\gamma\expe^{\iunit\theta}t}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t}{q}{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,q^{\frac{1}{2}},q^{\frac{1}{2}}\expe^{2\iunit\theta}}{q,\gamma\expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{q}{q}{n}}\frac{\ctsqLegendre{n}@{x}{q}}{q^{\frac{1}{4}n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary
Limit relations
Continuous q-Legendre polynomial to Legendre / Spherical polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\ctsqLegendre{n}@{x}{q}=\LegendrePoly{n}@{x} }}
Remarks
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqLegendre{n}@{x}{q}=q^{\frac{1}{4}n}\sum_{k=0}^n \frac{\qPochhammer{q^{\frac{1}{2}}}{q}{k}\qPochhammer{q^{\frac{1}{2}}}{q}{n-k}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}} \expe^{\iunit(n-2k)\theta} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqLegendreRahman{n}@{x}{q}=\qHyperrphis{4}{3}@@{q^{-n},q^{n+1},q^{\frac{1}{2}}\expe^{\iunit\theta},q^{\frac{1}{2}}\expe^{-\iunit\theta}}{q,-q,-q}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqLegendre{n}@{x}{q^2}=\ctsqLegendreRahman{n}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqLegendre{n}@{x}{q^{-1}}=\ctsqLegendre{n}@{x}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqLegendre{n}@{x}{q}=q^{\frac{1}{4}n}\ctsqUltra{n}@{x}{q^{\frac{1}{2}}}{q} }}