Continuous big q-Hermite

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Continuous big q-Hermite

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Basic hypergeometric representation

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

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

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

Orthogonality relation(s)

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

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

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

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

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

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w_{k}=\frac{\left(a^{-2};q\right)_{\infty}}{\left(q;q\right)_{\infty}}\frac{(1-a^{2}q^{2k})\left(a^{2};q\right)_{k}}{(1-a^{2})\left(q;q\right)_{k}}q^{-\frac{3}{2}k^{2}-\frac{1}{2}k}\left(-\frac{1}{a^{4}}\right)^{k}}}}$ &

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

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

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle 2xH_{n}\!\left(x;a|q\right)=H_{n+1}\!\left(x;a|q\right)+aq^{n}H_{n}\!\left(x;a|q\right)+(1-q^{n})H_{n-1}\!\left(x;a|q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{H}}_{n}\!\left(x\right)={\widehat{H}}_{n+1}\!\left(x\right)+\frac{1}{2}aq^{n}{\widehat{H}}_{n}\!\left(x\right)+\frac{1}{4}(1-q^{n}){\widehat{H}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x;a|q\right)=2^{n}{\widehat{H}}_{n}\!\left(x\right)}}}$

q-Difference equations

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

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

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

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

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

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

Forward shift operator

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

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

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

Backward shift operator

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

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

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

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

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

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

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

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

Limit relations

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

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

Continuous big q-Hermite polynomial to Continuous q-Hermite polynomial

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

Continuous big q-Hermite polynomial to Hermite polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{H_{n}\!\left(x\sqrt{\frac{1}{2}(1-q)};0|q\right)}{\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=H_{n}\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}\frac{H_{n}\!\left(x\sqrt{\frac{1}{2}(1-q)};a\sqrt{2(1-q)}|q\right)}{\left(\frac{1-q}{2}\right)^{\frac{n}{2}}}=H_{n}\left(x-a\right)}}}$

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Continuous big q-Hermite

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