Continuous q-Jacobi

From DRMF
Jump to navigation Jump to search

Continuous q-Jacobi

Basic hypergeometric representation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqJacobi{\alpha}{\beta}{n}@{x}{q} {}=\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}\ \qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+1},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{-\iunit\theta}} {q^{\alpha+1},-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{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)}{\sqrt{1-x^2}} \ctsqJacobi{\alpha}{\beta}{m}@{x}{q}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}\,dx {}=\frac{\qPochhammer{q^{\frac{1}{2}(\alpha+\beta+2)},q^{\frac{1}{2}(\alpha+\beta+3)}}{q}{\infty}}{\qPochhammer{q,q^{\alpha+1},q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+1)} -q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{\infty}}\,\frac{1-q^{\alpha+\beta+1}}{1-q^{2n+\alpha+\beta+1}} {}\frac{\qPochhammer{q^{\alpha+1},q^{\beta+1},-q^{\frac{1}{2}(\alpha+\beta+3)}}{q}{n}} {\qPochhammer{q,q^{\alpha+\beta+1},-q^{\frac{1}{2}(\alpha+\beta+1)}}{q}{n}}q^{(\alpha+\frac{1}{2})n}\,\Kronecker{m}{n} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta},-q^{\frac{1}{2}\beta+\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{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}}) h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}} &

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\normctsqJacobiPtilde{n}{\beta}{n}@@{x}{q}=A_n\normctsqJacobiPtilde{n+1}{\beta}{n}@@{x}{q}+\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+ q^{-\frac{1}{2}\alpha-\frac{1}{4}}-\left(A_n+C_n\right)\right]\normctsqJacobiPtilde{n}{\beta}{n}@@{x}{q} {}+C_n\normctsqJacobiPtilde{n-1}{\beta}{n}@@{x}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=\frac{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^n)(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})} {(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-q^{n+\alpha+1})(1-q^{n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(\alpha+\beta+2)})} {q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(1-q^{2n+\alpha+\beta+2})}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \normctsqJacobiPtilde{n}{\beta}{n}@@{x}{q}:=\normctsqJacobiPtilde{\alpha}{\beta}{n}@{x}{q}=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q} }}

Monic recurrence relation

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicctsqJacobi{\alpha}{\beta}{n}@@{x}{q}=\monicctsqJacobi{\alpha}{\beta}{n+1}@@{x}{q}+\frac{1}{2}\left[q^{\frac{1}{2}\alpha+\frac{1}{4}}+ q^{-\frac{1}{2}\alpha-\frac{1}{4}}-(A_n+C_n)\right]\monicctsqJacobi{\alpha}{\beta}{n}@@{x}{q} {}+\frac{1}{4}A_{n-1}C_n\monicctsqJacobi{\alpha}{\beta}{n-1}@@{x}{q} }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle C_n=\frac{q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^n)(1-q^{n+\beta})(1+q^{n+\frac{1}{2}(\alpha+\beta)})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})} {(1-q^{2n+\alpha+\beta})(1-q^{2n+\alpha+\beta+1})}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle A_n=\frac{(1-q^{n+\alpha+1})(1-q^{n+\alpha+\beta+1})(1+q^{n+\frac{1}{2}(\alpha+\beta+1)})(1+q^{n+\frac{1}{2}(\alpha+\beta+2)})} {q^{\frac{1}{2}\alpha+\frac{1}{4}}(1-q^{2n+\alpha+\beta+1})(1-q^{2n+\alpha+\beta+2})}}}


Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=\frac{2^nq^{(\frac{1}{2}\alpha+\frac{1}{4})n}\qPochhammer{q^{n+\alpha+\beta+1}}{q}{n}} {\qPochhammer{q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}}\monicctsqJacobi{\alpha}{\beta}{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^{\alpha+1},q^{\beta+1}|q)D_qy(x)\right]+ \lambda_n{\tilde w}(x;q^{\alpha},q^{\beta}|q)y(x)=0 }}

Substitution(s): 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+\alpha+\beta+1})}} &

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;q^{\alpha},q^{\beta}|q):=\frac{w(x;q^{\alpha},q^{\beta}|q)}{\sqrt{1-x^2}}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}}} &
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle w(x):=w(x;q^{\alpha},q^{\beta}|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta},-q^{\frac{1}{2}\beta+\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{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}}) h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}} &
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}}}


Forward shift operator

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_q\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=-\frac{q^{-n+\frac{1}{2}\alpha+\frac{3}{4}} (1-q^{n+\alpha+\beta+1})(\expe^{\iunit\theta}-\expe^{-\iunit\theta})}{(1+q^{\frac{1}{2}(\alpha+\beta+1)}) (1+q^{\frac{1}{2}(\alpha+\beta+2)})} {} P_{n-1}^{(\alpha+1,\beta+1)}(x|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\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=\frac{2q^{-n+\frac{1}{2}\alpha+\frac{5}{4}} (1-q^{n+\alpha+\beta+1})}{(1-q)(1+q^{\frac{1}{2}(\alpha+\beta+1)}) (1+q^{\frac{1}{2}(\alpha+\beta+2)})} {} P_{n-1}^{(\alpha+1,\beta+1)}(x|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;q^{\alpha},q^{\beta}|q)\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}\right] {}=q^{-\frac{1}{2}\alpha-\frac{1}{4}}(1-q^{n+1})(1+q^{\frac{1}{2}(\alpha+\beta-1)}) (1+q^{\frac{1}{2}(\alpha+\beta)})(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) {}{\tilde w}(x;q^{\alpha-1},q^{\beta-1}|q) P_{n+1}^{(\alpha-1,\beta-1)}(x|q) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;q^{\alpha},q^{\beta}|q):=\frac{w(x;q^{\alpha},q^{\beta}|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;q^{\alpha},q^{\beta}|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta},-q^{\frac{1}{2}\beta+\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{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}}) h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}} &
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;q^{\alpha},q^{\beta}|q)\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}\right] {}=-2q^{-\frac{1}{2}\alpha+\frac{1}{4}} \frac{(1-q^{n+1})(1+q^{\frac{1}{2}(\alpha+\beta-1)})(1+q^{\frac{1}{2}(\alpha+\beta)})}{1-q} {}{\tilde w}(x;q^{\alpha-1},q^{\beta-1}|q)P_{n+1}^{(\alpha-1,\beta-1)}(x|q) }}

Substitution(s): Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle {\tilde w}(x;q^{\alpha},q^{\beta}|q):=\frac{w(x;q^{\alpha},q^{\beta}|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;q^{\alpha},q^{\beta}|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta},-q^{\frac{1}{2}\beta+\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{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}}) h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}} &
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;q^\alpha,q^\beta|q)\ctsqJacobi{\alpha}{\beta}{n}@{x}{q} {}=\left(\frac{q-1}{2}\right)^n\frac{q^{\frac{1}{4}n^2+\frac{1}{2}n\alpha}} {\qPochhammer{q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}} {}\left(D_q\right)^n\left[{\tilde w}(x;q^{\alpha+n},q^{\beta+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;q^{\alpha},q^{\beta}|q):=\frac{w(x;q^{\alpha},q^{\beta}|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;q^{\alpha},q^{\beta}|q) =\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta},q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta},-q^{\frac{1}{2}\beta+\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{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta}}{q^{\frac{1}{2}}}{\infty}}\right|^2 =\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})} {h(x,q^{\frac{1}{2}\alpha+\frac{1}{4}})h(x,q^{\frac{1}{2}\alpha+\frac{3}{4}}) h(x,-q^{\frac{1}{2}\beta+\frac{1}{4}})h(x,-q^{\frac{1}{2}\beta+\frac{3}{4}})}}} &
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 \qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{\iunit\theta}}{q^{\alpha+1}}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{-q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{-\iunit\theta} -q^{\frac{1}{2}\beta+\frac{3}{4}}\expe^{-\iunit\theta}}{q^{\beta+1}}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}} {\qPochhammer{q^{\alpha+1},q^{\beta+1}}{q}{n}}\frac{\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})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 \qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{\iunit\theta}}{-q^{\frac{1}{2}(\alpha+\beta+1)}}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{-\iunit\theta} -q^{\frac{1}{2}\beta+\frac{3}{4}}\expe^{-\iunit\theta}}{-q^{\frac{1}{2}(\alpha+\beta+3)}}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}} {\qPochhammer{-q^{\frac{1}{2}(\alpha+\beta+3)}}{q}{n}}\frac{\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})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 \qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{1}{4}}\expe^{\iunit\theta} -q^{\frac{1}{2}\beta+\frac{3}{4}}\expe^{\iunit\theta}}{-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{\expe^{-\iunit\theta}t}\ \qHyperrphis{2}{1}@@{q^{\frac{1}{2}\alpha+\frac{3}{4}}\expe^{-\iunit\theta} -q^{\frac{1}{2}\beta+\frac{1}{4}}\expe^{-\iunit\theta}}{-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{\expe^{\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{-q^{\frac{1}{2}(\alpha+\beta+1)}}{q}{n}} {\qPochhammer{-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}}\frac{\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})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}}}


Limit relations

Askey-Wilson polynomial to Continuous q-Jacobi polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \frac{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}\AskeyWilson{n}@{x}{q^{\frac{1}{2}\alpha+\frac{1}{4}}}{q^{\frac{1}{2}\alpha+\frac{3}{4}}}{ -q^{\frac{1}{2}\beta+\frac{1}{4}}}{-q^{\frac{1}{2}\beta+\frac{3}{4}}}{q}} {\qPochhammer{q,-q^{\frac{1}{2}(\alpha+\beta+1)},-q^{\frac{1}{2}(\alpha+\beta+2)}}{q}{n}} =\ctsqJacobi{\alpha}{\beta}{n}@{x}{q} }}

Continuous q-Jacobi polynomial to Continuous q-Laguerre polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim _{\beta\rightarrow\infty} \ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=\ctsqLaguerre{\alpha}{n}@{x}{q} }}

Continuous q-Jacobi polynomial to Jacobi polynomial

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}=\Jacobi{\alpha}{\beta}{n}@{x} }}

Remarks

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqJacobiRahman{\alpha}{\beta}{n}@{x}{q}=\frac{\qPochhammer{q^{\alpha+1},-q^{\beta+1}}{q}{n}}{\qPochhammer{q,-q}{q}{n}} {} \qHyperrphis{4}{3}@@{q^{-n},q^{n+\alpha+\beta+1},q^{\frac{1}{2}}\expe^{\iunit\theta},q^{\frac{1}{2}}\expe^{-\iunit\theta}}{q^{\alpha+1},-q^{\beta+1},-q}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \ctsqJacobi{\alpha}{\beta}{n}@{x}{q^2}=\frac{\qPochhammer{-q}{q}{n}}{\qPochhammer{-q^{\alpha+\beta+1}}{q}{n}}q^{n\alpha}\ctsqJacobiRahman{\alpha}{\beta}{n}@{x}{q} }}
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 \ctsqJacobi{\alpha}{\beta}{n}@{x}{q^{-1}}=q^{-n\alpha}\ctsqJacobi{\alpha}{\beta}{n}@{x}{q}\quad\textrm{and}\quad \ctsqJacobiRahman{\alpha}{\beta}{n}@{x}{q^{-1}}=q^{-n(\alpha+\beta)}\ctsqJacobiRahman{\alpha}{\beta}{n}@{x}{q} }}

Koornwinder Addendum: q-Meixner-Pollaczek

Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qMeixnerPollaczek{n}@{x}{a }{ q}:=\frac1{\qPochhammer{q}{q}{n}} \AskeyWilson{n}@{x}{a \expe^{\iunit\phi}}{0}{a \expe^{-\iunit\phi}}{0 }{ q} (x=\cos@{\theta+\phi}) }}

Retrieved from "https://drmf-beta.wmflabs.org/index.php?title=Continuous_q-Jacobi&oldid=515"