Al-Salam-Chihara

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Al-Salam-Chihara

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=\frac{\left(ab;q\right)_{n}}{a^{n}}\ \qHyperrphis{3}{2}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab,0}{q}{q}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=\left(a{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{n}{\mathrm{e}^{-\mathrm{i}n\theta}}\,\qHyperrphis{2}{1}@@{q^{-n},b{\mathrm{e}^{-\mathrm{i}\theta}}}{a^{-1}q^{-n+1}{\mathrm{e}^{-\mathrm{i}\theta}}}{q}{a^{-1}q{\mathrm{e}^{\mathrm{i}\theta}}}}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=\left(b{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{n}{\mathrm{e}^{\mathrm{i}n\theta}}\,\qHyperrphis{2}{1}@@{q^{-n},a{\mathrm{e}^{\mathrm{i}\theta}}}{b^{-1}q^{-n+1}{\mathrm{e}^{\mathrm{i}\theta}}}{q}{b^{-1}q{\mathrm{e}^{-\mathrm{i}\theta}}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x)}{\sqrt{1-x^{2}}}Q_{m}\!\left(x;a,b\,|\,q\right)Q_{n}\!\left(x;a,b\,|\,q\right)\,dx=\frac{\,\delta_{m,n}}{\left(q^{n+1},abq^{n};q\right)_{\infty}}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)}}}}$ &

${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{-1}^{1}\frac{w(x)}{\sqrt{1-x^{2}}}Q_{m}\!\left(x;a,b\,|\,q\right)Q_{n}\!\left(x;a,b\,|\,q\right)\,dx{}+\sum_{\begin{array}[]{c}\scriptstyle k\\ \scriptstyle 1<aq^{k}\leq a\end{array}}w_{k}Q_{m}\!\left(x_{k};a,b\,|\,q\right)Q_{n}\!\left(x_{k};a,b\,|\,q\right)=\frac{\,\delta_{m,n}}{\left(q^{n+1},abq^{n};q\right)_{\infty}}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w_{k}=\frac{\left(a^{-2};q\right)_{\infty}}{\left(q,ab,a^{-1}b;q\right)_{\infty}}\frac{(1-a^{2}q^{2k})\left(a^{2},ab;q\right)_{k}}{(1-a^{2})\left(q,ab^{-1}q;q\right)_{k}}q^{-k^{2}}\left(\frac{1}{a^{3}b}\right)^{k}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x_{k}=\frac{aq^{k}+\left(aq^{k}\right)^{-1}}{2}}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle 2xQ_{n}\!\left(x\right)=Q_{n+1}\!\left(x\right)+(a+b)q^{n}Q_{n}\!\left(x\right)+(1-q^{n})(1-abq^{n-1})Q_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x\right):=Q_{n}\!\left(x;a,b\,|\,q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x;a,b\,|\,q\right)={\widehat{Q}}_{n+1}\!\left(x;a,b\,|\,q\right)+\frac{1}{2}(a+b)q^{n}{\widehat{Q}}_{n}\!\left(x;a,b\,|\,q\right)+\frac{1}{4}(1-q^{n})(1-abq^{n-1}){\widehat{Q}}_{n-1}\!\left(x;a,b\,|\,q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,|\,q\right)=2^{n}{\widehat{Q}}_{n}\!\left(x;a,b\,|\,q\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle(1-q)^{2}D_{q}\left[{\tilde{w}}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}|q)D_{q}y(x)\right]{}+4q^{-n+1}(1-q^{n}){\tilde{w}}(x;a,b|q)y(x)=0}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b|q):=\frac{w(x;a,b|q)}{\sqrt{1-x^{2}}}}}}$ &

${\displaystyle{\displaystyle{\displaystyle y(x)=Q_{n}\!\left(x;a,b\,|\,q\right)}}}$ &
${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

${\displaystyle{\displaystyle{\displaystyle\LegendrePoly{n}@{z}:=\frac{\left(ab;q\right)_{n}}{a^{n}}\,\qHyperrphis{3}{2}@@{q^{-n},az,az^{-1}}{ab,0}{q}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle q^{-n}(1-q^{n})\LegendrePoly{n}@{z}=A(z)\LegendrePoly{n}@{qz}-\left[A(z)+A(z^{-1})\right]\LegendrePoly{n}@{z}+A(z^{-1})\LegendrePoly{n}@{q^{-1}z}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle A(z)=\frac{(1-az)(1-bz)}{(1-z^{2})(1-qz^{2})}}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}Q_{n}\!\left(x;a,b\,|\,q\right){}=-q^{-\frac{1}{2}n}(1-q^{n})({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}})Q_{n-1}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}|q)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

${\displaystyle{\displaystyle{\displaystyle D_{q}Q_{n}\!\left(x;a,b\,|\,q\right)=2q^{-\frac{1}{2}(n-1)}\frac{1-q^{n}}{1-q}Q_{n-1}(x;aq^{\frac{1}{2}},bq^{\frac{1}{2}}|q)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle\delta_{q}\left[{\tilde{w}}(x;a,b|q)Q_{n}\!\left(x;a,b\,|\,q\right)\right]{}=q^{-\frac{1}{2}(n+1)}({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}\theta}}){\tilde{w}}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q){}Q_{n+1}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b|q):=\frac{w(x;a,b|q)}{\sqrt{1-x^{2}}}}}}$ &

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

${\displaystyle{\displaystyle{\displaystyle D_{q}\left[{\tilde{w}}(x;a,b|q)Q_{n}\!\left(x;a,b\,|\,q\right)\right]{}=-\frac{2q^{-\frac{1}{2}n}}{1-q}{\tilde{w}}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q)Q_{n+1}(x;aq^{-\frac{1}{2}},bq^{-\frac{1}{2}}|q)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b|q):=\frac{w(x;a,b|q)}{\sqrt{1-x^{2}}}}}}$ &

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b|q)Q_{n}\!\left(x;a,b\,|\,q\right){}=\left(\frac{q-1}{2}\right)^{n}q^{\frac{1}{4}n(n-1)}\left(D_{q}\right)^{n}\left[{\tilde{w}}(x;aq^{\frac{1}{2}n},bq^{\frac{1}{2}n}|q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\tilde{w}}(x;a,b|q):=\frac{w(x;a,b|q)}{\sqrt{1-x^{2}}}}}}$ &

${\displaystyle{\displaystyle{\displaystyle w(x):=w(x;a,b|q)=\left|\frac{\left({\mathrm{e}^{2\mathrm{i}\theta}};q\right)_{\infty}}{\left(a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}};q\right)_{\infty}}\right|^{2}=\frac{h(x,1)h(x,-1)h(x,q^{\frac{1}{2}})h(x,-q^{\frac{1}{2}})}{h(x,a)h(x,b)}}}}$ &
${\displaystyle{\displaystyle{\displaystyle h(x,\alpha):=\prod_{k=0}^{\infty}\left(1-2\alpha xq^{k}+\alpha^{2}q^{2k}\right)=\left(\alpha{\mathrm{e}^{\mathrm{i}\theta}},\alpha{\mathrm{e}^{-\mathrm{i}\theta}};q\right)_{\infty}}}}$ &

${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{\left(at,bt;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t,{\mathrm{e}^{-\mathrm{i}\theta}}t;q\right)_{\infty}}=\sum_{n=0}^{\infty}\frac{Q_{n}\!\left(x;a,b\,|\,q\right)}{\left(q;q\right)_{n}}t^{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left({\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}\ \qHyperrphis{2}{1}@@{a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}}{ab}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}=\sum_{n=0}^{\infty}\frac{Q_{n}\!\left(x;a,b\,|\,q\right)}{\left(ab,q;q\right)_{n}}t^{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

${\displaystyle{\displaystyle{\displaystyle\left(t;q\right)_{\infty}\cdot\qHyperrphis{2}{1}@@{a{\mathrm{e}^{\mathrm{i}\theta}},a{\mathrm{e}^{-\mathrm{i}\theta}}}{ab}{q}{t}{}=\sum_{n=0}^{\infty}\frac{(-1)^{n}a^{n}q^{\binomial{n}{2}}}{\left(ab,q;q\right)_{n}}Q_{n}\!\left(x;a,b\,|\,q\right)t^{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$

${\displaystyle{\displaystyle{\displaystyle\frac{\left(\gamma{\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}{\left({\mathrm{e}^{\mathrm{i}\theta}}t;q\right)_{\infty}}\ \qHyperrphis{3}{2}@@{\gamma,a{\mathrm{e}^{\mathrm{i}\theta}},b{\mathrm{e}^{\mathrm{i}\theta}}}{ab,\gamma{\mathrm{e}^{\mathrm{i}\theta}}t}{q}{{\mathrm{e}^{-\mathrm{i}\theta}}t}{}=\sum_{n=0}^{\infty}\frac{\left(\gamma;q\right)_{n}}{\left(ab,q;q\right)_{n}}Q_{n}\!\left(x;a,b\,|\,q\right)t^{n}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}$ &
${\displaystyle{\displaystyle{\displaystyle\gamma}}}$ arbitrary

Limit relations

Continuous dual q-Hahn polynomial to Al-Salam-Chihara polynomial

${\displaystyle{\displaystyle{\displaystyle p_{n}\!\left(x;a,b,0|q\right)=Q_{n}\!\left(x;a,b\,|\,q\right)}}}$

Al-Salam-Chihara polynomial to Continuous big q-Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,0\,|\,q\right)=H_{n}\!\left(x;a|q\right)}}}$

Al-Salam-Chihara polynomial to Continuous q-Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;q^{\frac{1}{2}\alpha+\frac{1}{4}},q^{\frac{1}{2}\alpha+\frac{3}{4}}\,|\,q\right)=\frac{\left(q;q\right)_{n}}{q^{(\frac{1}{2}\alpha+\frac{1}{4})n}}P^{(\alpha)}_{n}\!\left(x|q\right)}}}$

Al-Salam-Chihara polynomial to Meixner-Pollaczek polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{Q_{n}\!\left(\cos\left(\ln q^{x}+\phi\right);q^{\lambda}{\mathrm{e}^{\mathrm{i}\phi}},q^{\lambda}{\mathrm{e}^{-\mathrm{i}\phi}}\,|\,q\right)}{\left(q;q\right)_{n}}=P^{(\lambda)}_{n}\!\left(x;\phi\right)}}}$

Koornwinder Addendum: Al-Salam-Chihara

Re: (14.8.1)

${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}\right)=(-1)^{n}b^{n}q^{-\frac{1}{2}n(n-1)}\left((ab)^{-1};q\right)_{n}\qHyperrphis{3}{1}@@{q^{-n},q^{-x},a^{-2}q^{x}}{(ab)^{-1}}{q}{q^{n}ab^{-1}}}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}\right)=(-ab^{-1})^{x}q^{-\frac{1}{2}x(x+1)}\frac{\left(qba^{-1};q\right)_{x}}{\left(a^{-1}b^{-1};q\right)_{x}}\qHyperrphis{2}{1}@@{q^{-x},a^{-2}q^{x}}{qba^{-1}}{q}{q^{n+1}}}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}\right)=(-ab^{-1})^{x}q^{-\frac{1}{2}x(x+1)}\frac{\left(qba^{-1};q\right)_{x}}{\left(a^{-1}b^{-1};q\right)_{x}}p_{x}\!\left(q^{n};ba^{-1},(qab)^{-1};q\right)}}}$

Orthogonality

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{(1-q^{2x}a^{-2})\left(a^{-2},(ab)^{-1};q\right)_{x}}{(1-a^{-2})\left(q,bqa^{-1};q\right)_{x}}(ba^{-1})^{x}q^{x^{2}}Q_{m}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}\right)Q_{n}\!\left(\frac{1}{2}(aq^{-x}+a^{-1}q^{x});a,b\,;\,q^{-1}\right)=\frac{\left(qa^{-2};q\right)_{\infty}}{\left(ba^{-1}q;q\right)_{\infty}}\left(q,(ab)^{-1};q\right)_{n}(ab)^{n}q^{-n^{2}}\delta_{m,n}(ab>1,\;qb<a)}}}$

Normalized recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{Q}}_{n}\!\left(x\right)={\widehat{Q}}_{n+1}\!\left(x\right)+\frac{1}{2}(a+b)q^{-n}{\widehat{Q}}_{n}\!\left(x\right)+\tfrac{1}{4}(q^{-n}-1)(abq^{-n+1}-1){\widehat{Q}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle Q_{n}\!\left(x;a,b\,;\,q^{-1}\right)=2^{n}{\widehat{Q}}_{n}\!\left(x\right)}}}$

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Al-Salam-Chihara

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