Al-Salam-Chihara
Al-Salam-Chihara
Basic hypergeometric representation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}=\frac{\qPochhammer{ab}{q}{n}}{a^n}\ \qHyperrphis{3}{2}@@{q^{-n},a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{ab,0}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}=\qPochhammer{a\expe^{\iunit\theta}}{q}{n}\expe^{-\iunit n\theta}\,\qHyperrphis{2}{1}@@{q^{-n},b\expe^{-\iunit\theta}}{a^{-1}q^{-n+1}\expe^{-\iunit\theta}}{q}{a^{-1}q\expe^{\iunit\theta}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}=\qPochhammer{b\expe^{-\iunit\theta}}{q}{n}\expe^{\iunit n\theta}\,\qHyperrphis{2}{1}@@{q^{-n},a\expe^{\iunit\theta}}{b^{-1}q^{-n+1}\expe^{\iunit\theta}}{q}{b^{-1}q\expe^{-\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}}\AlSalamChihara{m}@{x}{a}{b}{q}\AlSalamChihara{n}@{x}{a}{b}{q}\,dx =\frac{\,\Kronecker{m}{n}}{\qPochhammer{q^{n+1},abq^n}{q}{\infty}} }}
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 \frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\AlSalamChihara{m}@{x}{a}{b}{q}\AlSalamChihara{n}@{x}{a}{b}{q}\,dx {}+\sum_{\begin{array}{c}\scriptstyle k\\ \scriptstyle 1<aq^k\leq a\end{array}} w_k\AlSalamChihara{m}@{x_k}{a}{b}{q}\AlSalamChihara{n}@{x_k}{a}{b}{q}=\frac{\,\Kronecker{m}{n}}{\qPochhammer{q^{n+1},abq^n}{q}{\infty}} }}
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}}}
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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|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\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)}}}
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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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Recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle 2x\AlSalamChihara{n}@@{x}{a}{b}{q}=\AlSalamChihara{n+1}@@{x}{a}{b}{q}+(a+b)q^n\AlSalamChihara{n}@@{x}{a}{b}{q}+(1-q^n)(1-abq^{n-1})\AlSalamChihara{n-1}@@{x}{a}{b}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamChihara{n}@@{x}{a}{b}{q}:=\AlSalamChihara{n}@{x}{a}{b}{q} }}
Monic recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicAlSalamChihara{n}@@{x}{a}{b}{q}=\monicAlSalamChihara{n+1}@@{x}{a}{b}{q}+\frac{1}{2}(a+b)q^n\monicAlSalamChihara{n}@@{x}{a}{b}{q}+\frac{1}{4}(1-q^n)(1-abq^{n-1})\monicAlSalamChihara{n-1}@@{x}{a}{b}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamChihara{n}@{x}{a}{b}{q}=2^n\monicAlSalamChihara{n}@@{x}{a}{b}{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}}|q)D_qy(x)\right] {}+4q^{-n+1}(1-q^n){\tilde w}(x;a,b|q)y(x)=0 }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle y(x)=\AlSalamChihara{n}@{x}{a}{b}{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;a,b|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\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)}}}
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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 \LegendrePoly{n}@{z}:=\frac{\qPochhammer{ab}{q}{n}}{a^n}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{ab,0}{q}{q} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle q^{-n}(1-q^n)\LegendrePoly{n}@{z}=A(z)\LegendrePoly{n}@{qz}-\left[A(z)+A(z^{-1})\right]\LegendrePoly{n}@{z} +A(z^{-1})\LegendrePoly{n}@{q^{-1}z} }}
Forward shift operator
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \delta_q\AlSalamChihara{n}@{x}{a}{b}{q} {}=-q^{-\frac{1}{2}n}(1-q^n)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) Q_{n-1}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}|q) }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle D_q\AlSalamChihara{n}@{x}{a}{b}{q}=2q^{-\frac{1}{2}(n-1)} \frac{1-q^n}{1-q}Q_{n-1}(x;aq^{\frac{1}{2}},bq^{\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|q)\AlSalamChihara{n}@{x}{a}{b}{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}}|q) {} Q_{n+1}(x;aq^{-\frac{1}{2}},bq^{-\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|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\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)}}}
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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;a,b|q)\AlSalamChihara{n}@{x}{a}{b}{q}\right] {}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde w}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q) Q_{n+1}(x;aq^{-\frac{1}{2}},bq^{-\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|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\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)}}}
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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;a,b|q)\AlSalamChihara{n}@{x}{a}{b}{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}|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|q)=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}} {\qPochhammer{a\expe^{\iunit\theta},b\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)}}}
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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 \frac{\qPochhammer{at,bt}{q}{\infty}}{\qPochhammer{\expe^{\iunit\theta}t,\expe^{-\iunit\theta}t}{q}{\infty}} =\sum_{n=0}^{\infty}\frac{\AlSalamChihara{n}@{x}{a}{b}{q}}{\qPochhammer{q}{q}{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}@@{a\expe^{\iunit\theta},b\expe^{\iunit\theta}}{ab}{q}{\expe^{-\iunit\theta}t} =\sum_{n=0}^{\infty}\frac{\AlSalamChihara{n}@{x}{a}{b}{q}}{\qPochhammer{ab,q}{q}{n}}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qPochhammer{t}{q}{\infty}\cdot\qHyperrphis{2}{1}@@{a\expe^{\iunit\theta},a\expe^{-\iunit\theta}}{ab}{q}{t} {}=\sum_{n=0}^{\infty}\frac{(-1)^na^nq^{\binomial{n}{2}}}{\qPochhammer{ab,q}{q}{n}}\AlSalamChihara{n}@{x}{a}{b}{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,a\expe^{\iunit\theta},b\expe^{\iunit\theta}}{ab,\gamma \expe^{\iunit\theta}t}{q}{\expe^{-\iunit\theta}t} {}=\sum_{n=0}^{\infty}\frac{\qPochhammer{\gamma}{q}{n}}{\qPochhammer{ab,q}{q}{n}}\AlSalamChihara{n}@{x}{a}{b}{q}t^n }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \gamma}} arbitrary
Limit relations
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} }}
Al-Salam-Chihara polynomial to Continuous big q-Hermite polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamChihara{n}@{x}{a}{0}{q}=\ctsbigqHermite{n}@{x}{a}{q} }}
Al-Salam-Chihara polynomial to Continuous q-Laguerre polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \AlSalamChihara{n}@{x}{q^{\frac{1}{2}\alpha+\frac{1}{4}}}{q^{\frac{1}{2}\alpha+\frac{3}{4}}}{q} =\frac{\qPochhammer{q}{q}{n}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}\ctsqLaguerre{\alpha}{n}@{x}{q} }}
Al-Salam-Chihara polynomial to Meixner-Pollaczek polynomial
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \lim_{q\rightarrow 1}\frac{\AlSalamChihara{n}@{\cos@{\ln@@{q^x}+\phi}}{ q^{\lambda}\expe^{\iunit\phi}}{q^{\lambda}\expe^{-\iunit\phi}}{q}}{\qPochhammer{q}{q}{n}}=\MeixnerPollaczek{\lambda}{n}@{x}{\phi} }}
Koornwinder Addendum: Al-Salam-Chihara
Re: (14.8.1)
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qinvAlSalamChihara{n}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}}= (-1)^n b^n q^{-\frac12 n(n-1)}\qPochhammer{(ab)^{-1}}{q}{n} \qHyperrphis{3}{1}@@{q^{-n},q^{-x},a^{-2}q^x}{(ab)^{-1}}{q}{q^nab^{-1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qinvAlSalamChihara{n}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}} =(-ab^{-1})^x q^{-\frac12 x(x+1)} \frac{\qPochhammer{qba^{-1}}{q}{x}}{\qPochhammer{a^{-1}b^{-1}}{q}{x}} \qHyperrphis{2}{1}@@{q^{-x},a^{-2}q^x}{qba^{-1}}{q}{q^{n+1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qinvAlSalamChihara{n}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}} =(-ab^{-1})^x q^{-\frac12 x(x+1)} \frac{\qPochhammer{qba^{-1}}{q}{x}}{\qPochhammer{a^{-1}b^{-1}}{q}{x}} \littleqJacobi{x}@{q^n}{ba^{-1}}{(qab)^{-1}}{q} }}
Orthogonality
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \sum_{x=0}^\infty \frac{(1-q^{2x}a^{-2}) \qPochhammer{a^{-2},(ab)^{-1}}{q}{x}} {(1-a^{-2}) \qPochhammer{q,bqa^{-1}}{q}{x}} (ba^{-1})^xq^{x^2} \qinvAlSalamChihara{m}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}}\qinvAlSalamChihara{n}@{\frac12(aq^{-x}+a^{-1}q^x)}{a}{b }{q^{-1}} =\frac{\qPochhammer{qa^{-2}}{q}{\infty}}{\qPochhammer{ba^{-1}q}{q}{\infty}} \qPochhammer{q,(ab)^{-1}}{q}{n} (ab)^nq^{-n^2} \Kronecker{m}{n} (ab>1,\;qb<a) }}
Normalized recurrence relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle x\monicqinvAlSalamChihara{n}@@{x}{a}{b }{q^{-1}}=\monicqinvAlSalamChihara{n+1}@@{x}{a}{b }{q^{-1}}+\frac12(a+b)q^{-n} \monicqinvAlSalamChihara{n}@@{x}{a}{b }{q^{-1}}+ \tfrac14(q^{-n}-1)(abq^{-n+1}-1)\monicqinvAlSalamChihara{n-1}@@{x}{a}{b }{q^{-1}} }}
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \qinvAlSalamChihara{n}@{x}{a}{b }{q^{-1}}=2^n \monicqinvAlSalamChihara{n}@@{x}{a}{b }{q^{-1}} }}