Continuous q-Hahn
Continuous q-Hahn
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{(a\expe^{\iunit\phi})^n\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab\expe^{2\iunit\phi},ac,ad}{q}{n}} {}=\qHyperrphis{4}{3}@@{q^{-n},abcdq^{n-1},a\expe^{\iunit(\theta+2\phi)},a\expe^{-\iunit\theta}}{ab\expe^{2\iunit\phi},ac,ad}{q}{q} }}
Orthogonality relation(s)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{1}{4\cpi}\int_{-\cpi}^{\cpi}w(\cos@{\theta+\phi}) \ctsqHahn{m}@{\cos@{\theta+\phi}}{a}{b}{c}{d}{q}\ctsqHahn{n}@{\cos@{\theta+\phi}}{a}{b}{c}{d}{q}\,d\theta {}=\frac{\qPochhammer{abcdq^{n-1}}{q}{n}\qPochhammer{abcdq^{2n}}{q}{\infty}} {\qPochhammer{q^{n+1},abq^n\expe^{2\iunit\phi},acq^n,adq^n,bcq^n,bdq^n,cdq^n\expe^{-2\iunit\phi}}{q}{\infty}}\,\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+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{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 2x\normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q}=A_n\normctsqHahnptilde{n+1}@@{x}{a}{b}{c}{d}{q}+\left[a\expe^{\iunit\phi}+a^{-1}\expe^{-\iunit\phi}-\left(A_n+C_n\right)\right]\normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q}+C_n\normctsqHahnptilde{n-1}@@{x}{a}{b}{c}{d}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-ab\expe^{2\iunit\phi}q^n)(1-acq^n)(1-adq^n)(1-abcdq^{n-1})}{a\expe^{\iunit\phi}(1-abcdq^{2n-1})(1-abcdq^{2n})}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \normctsqHahnptilde{n}@@{x}{a}{b}{c}{d}{q}:=\normctsqHahnptilde{n}@{x}{a}{b}{c}{d}{q}=\frac{(a\expe^{\iunit\phi})^n\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ab\expe^{2\iunit\phi},ac,ad}{q}{n}} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q}=\monicctsqHahn{n+1}@@{x}{a}{b}{c}{d}{q}+\frac{1}{2}\left[a\expe^{\iunit\phi}+a^{-1}\expe^{-\iunit\phi}-(A_n+C_n)\right]\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{q} {}+\frac{1}{4}A_{n-1}C_n\monicctsqHahn{n-1}@@{x}{a}{b}{c}{d}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-ab\expe^{2\iunit\phi}q^n)(1-acq^n)(1-adq^n)(1-abcdq^{n-1})}{a\expe^{\iunit\phi}(1-abcdq^{2n-1})(1-abcdq^{2n})}}}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqHahn{n}@{x}{a}{b}{c}{d}{q}=2^n\qPochhammer{abcdq^{n-1}}{q}{n}\monicctsqHahn{n}@@{x}{a}{b}{c}{d}{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}},dq^{\frac{1}{2}};q)D_qy(x)\right] {}+\lambda_n{\tilde w}(x;a,b,c,d;q)y(x)=0 }}
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-abcdq^{n-1})}}
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Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;a,b,c,d;q):=\frac{w(x;a,b,c,d;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 w(x):=w(x;a,b,c,d;q) =\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},b\expe^{\iunit(\theta+2\phi)} c\expe^{\iunit\theta},d\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\expe^{\iunit\phi})h(x,b\expe^{\iunit\phi})h(x,c\expe^{-\iunit\phi})h(x,d\expe^{-\iunit\phi})}}}
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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+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{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 \ctsqHahn{n}@{x}{a}{b}{c}{d}{q} {}=-q^{-\frac{1}{2}n}(1-q^n)(1-abcdq^{n-1})(\expe^{\iunit(\theta+\phi)}-\expe^{-\iunit(\theta+\phi)}) {} \ctsqHahn{n-1}@{x}{aq^{\frac{1}{2}}}{bq^{\frac{1}{2}}}{cq^{\frac{1}{2}}}{dq^{\frac{1}{2}}}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D_q \ctsqHahn{n}@{x}{a}{b}{c}{d}{q}=2q^{-\frac{1}{2}(n-1)}\frac{(1-q^n)(1-abcdq^{n-1})}{1-q} {} \ctsqHahn{n-1}@{x}{aq^{\frac{1}{2}}}{bq^{\frac{1}{2}}}{cq^{\frac{1}{2}}}{dq^{\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,d;q)\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}\right] {}=q^{-\frac{1}{2}(n+1)}(\expe^{\iunit(\theta+\phi)}-\expe^{-\iunit(\theta+\phi)}) {\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}},cq^{-\frac{1}{2}},dq^{-\frac{1}{2}};q) {} \ctsqHahn{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{dq^{-\frac{1}{2}}}{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,d;q) =\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},b\expe^{\iunit(\theta+2\phi)} c\expe^{\iunit\theta},d\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\expe^{\iunit\phi})h(x,b\expe^{\iunit\phi})h(x,c\expe^{-\iunit\phi})h(x,d\expe^{-\iunit\phi})}}}
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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+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{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;a,b,c,d;q)\ctsqHahn{n}@{x}{a}{b}{c}{d}{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}},dq^{-\frac{1}{2}};q) {} \ctsqHahn{n+1}@{x}{aq^{-\frac{1}{2}}}{bq^{-\frac{1}{2}}}{cq^{-\frac{1}{2}}}{dq^{-\frac{1}{2}}}{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,d;q) =\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},b\expe^{\iunit(\theta+2\phi)} c\expe^{\iunit\theta},d\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\expe^{\iunit\phi})h(x,b\expe^{\iunit\phi})h(x,c\expe^{-\iunit\phi})h(x,d\expe^{-\iunit\phi})}}}
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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+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{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;a,b,c,d;q)\ctsqHahn{n}@{x}{a}{b}{c}{d}{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},dq^{\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):=w(x;a,b,c,d;q) =\left|\frac{\qPochhammer{\expe^{2\iunit(\theta+\phi)}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit(\theta+2\phi)},b\expe^{\iunit(\theta+2\phi)} c\expe^{\iunit\theta},d\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\expe^{\iunit\phi})h(x,b\expe^{\iunit\phi})h(x,c\expe^{-\iunit\phi})h(x,d\expe^{-\iunit\phi})}}}
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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+\phi)},\alpha\expe^{-\iunit(\theta+\phi)}}{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 \qHyperrphis{2}{1}@@{a\expe^{\iunit(\theta+2\phi)},b\expe^{\iunit(\theta+2\phi)}}{ab\expe^{2\iunit\phi}}{q}{\expe^{-\iunit(\theta+\phi)}t} {} \qHyperrphis{2}{1}@@{c\expe^{-\iunit(\theta+2\phi)},d\expe^{-\iunit(\theta+2\phi)}}{cd\expe^{-2\iunit\phi}}{q}{\expe^{\iunit(\theta+\phi)}t} {}=\sum_{n=0}^{\infty}\frac{\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}t^n}{\qPochhammer{ab\expe^{2\iunit\phi},cd\expe^{-2\iunit\phi},q}{q}{n}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit(\theta+2\phi)},c\expe^{\iunit\theta}}{ac}{q}{\expe^{-\iunit(\theta+\phi)}t}\ \qHyperrphis{2}{1}@@{b\expe^{-\iunit\theta},d\expe^{-\iunit(\theta+2\phi)}}{bd}{q}{\expe^{\iunit(\theta+\phi)}t} {}=\sum_{n=0}^{\infty}\frac{\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ac,bd,q}{q}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qHyperrphis{2}{1}@@{a\expe^{\iunit(\theta+2\phi)},d\expe^{\iunit\theta}}{ad}{q}{\expe^{-\iunit(\theta+\phi)}t}\ \qHyperrphis{2}{1}@@{b\expe^{-\iunit\theta},c\expe^{-\iunit(\theta+2\phi)}}{bc}{q}{\expe^{\iunit(\theta+\phi)}t} {}=\sum_{n=0}^{\infty}\frac{\ctsqHahn{n}@{x}{a}{b}{c}{d}{q}}{\qPochhammer{ad,bc,q}{q}{n}}t^n }}
Limit relations
Askey-Wilson polynomial to Continuous q-Hahn polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AskeyWilson{n}@{\cos@{\theta+\phi}}{a\expe^{\iunit\phi}}{b\expe^{\iunit\phi}}{c\expe^{-\iunit\phi}}{d\expe^{-\iunit\phi}}{q}=\ctsqHahn{n}@{\cos@{\theta+\phi}}{a}{b}{c}{d}{q} }}
Continuous q-Hahn polynomial to q-Meixner-Pollaczek polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{\ctsqHahn{n}@{\cos@{\theta+\phi}}{a}{0}{0}{a}{q}}{\qPochhammer{q}{q}{n}}=\qMeixnerPollaczek{n}@{\cos@{\theta+\phi}}{a}{q} }}
Continuous q-Hahn polynomial to Continuous Hahn polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\frac{\ctsqHahn{n}@{\cos@{\ln@@{q^{-x}}+\phi}}{q^a}{q^b}{q^c}{q^d}{q}} {(1-q)^n\qPochhammer{q}{q}{n}}=(-2\sin@@{\phi})^n\ctsHahn{n}@{x}{a}{b}{c}{d} }}
Remark
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \normctsqHahnptilde{n}@{x}{a}{b}{c}{d}{q^{-1}}=\normctsqHahnptilde{n}@{x}{a^{-1}}{b^{-1}}{c^{-1}}{d^{-1}}{q} }}
Koornwinder Addendum: Continuous q-Hahn
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \littleqJacobi{n}@{x}{a}{b }{ q}:=\AskeyWilson{n}@{x}{a \expe^{\iunit\phi}}{b \expe^{\iunit\phi}}{a \expe^{-\iunit\phi}}{b \expe^{-\iunit\phi} }{ q} (x=\cos@{\theta+\phi}) }}