Continuous q-Hermite

<< Al-Salam-Carlitz II

Continuous q-Hermite

Stieltjes-Wigert >>

Basic hypergeometric representation

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

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

Orthogonality relation(s)

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

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

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

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

Recurrence relation

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

Monic recurrence relation

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

q-Difference equation

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

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

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

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

Forward shift operator

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

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

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

Backward shift operator

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

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

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

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

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

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

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

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

Rodrigues-type formula

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

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

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

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

Generating functions

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

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

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

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

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

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

Limit relations

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 q-Laguerre polynomial to Continuous q-Hermite polynomial

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

Continuous q-Hermite polynomial to Hermite polynomial

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

Remark

${\displaystyle{\displaystyle{\displaystyle H_{n}\!\left(x\,|\,q\right)=\sum_{k=0}^{n}\frac{\left(q;q\right)_{n}}{\left(q;q\right)_{k}\left(q;q\right)_{n-k}}{\mathrm{e}^{\mathrm{i}(n-2k)\theta}}}}}$

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

<< Al-Salam-Carlitz II

Continuous q-Hermite

Stieltjes-Wigert >>