Q-Charlier

<< q-Bessel

q-Charlier

Al-Salam-Carlitz I >>

Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(q^{-x};a;q\right)=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{0}{q}{-\frac{q^{n+1}}{a}}}}}$
${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(q^{-x};a;q\right)=\left(-a^{-1}q;q\right)_{n}\cdot\qHyperrphis{1}{1}@@{q^{-n}}{-a^{-1}q}{q}{-\frac{q^{n+1-x}}{a}}}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{x=0}^{\infty}\frac{a^{x}}{\left(q;q\right)_{x}}q^{\binomial{x}{2}}C_{m}\!\left(q^{-x};a;q\right)C_{n}\!\left(q^{-x};a;q\right){}=q^{-n}\left(-a;q\right)_{\infty}\left(-a^{-1}q,q;q\right)_{n}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a>0}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle q^{2n+1}(1-q^{-x})C_{n}\!\left(q^{-x}\right){}=aC_{n+1}\!\left(q^{-x}\right)-\left[a+q(1-q^{n})(a+q^{n})\right]C_{n}\!\left(q^{-x}\right){}+q(1-q^{n})(a+q^{n})C_{n-1}\!\left(q^{-x}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(q^{-x}\right):=C_{n}\!\left(q^{-x};a;q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{C}}_{n}\!\left(x\right)={\widehat{C}}_{n+1}\!\left(x\right)+\left[1+q^{-2n-1}\left\{a+q(1-q^{n})(a+q^{n})\right\}\right]{\widehat{C}}_{n}\!\left(x\right){}+aq^{-4n+1}(1-q^{n})(a+q^{n}){\widehat{C}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(q^{-x};a;q\right)=\frac{(-1)^{n}q^{n^{2}}}{a^{n}}{\widehat{C}}_{n}\!\left(q^{-x}\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle q^{n}y(x)=aq^{x}y(x+1)-q^{x}(a-1)y(x)+(1-q^{x})y(x-1)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle y(x)=C_{n}\!\left(q^{-x};a;q\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(q^{-x-1};a;q\right)-C_{n}\!\left(q^{-x};a;q\right)=-a^{-1}q^{-x}(1-q^{n})C_{n-1}\!\left(q^{-x};aq^{-1};q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\Delta C_{n}\!\left(q^{-x};a;q\right)}{\Delta q^{-x}}=-\frac{q(1-q^{n})}{a(1-q)}C_{n-1}\!\left(q^{-x};aq^{-1};q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle C_{n}\!\left(q^{-x};a;q\right)-a^{-1}q^{-x}(1-q^{x})C_{n}\!\left(q^{-x+1};a;q\right)=C_{n+1}\!\left(q^{-x};aq;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;a;q)C_{n}\!\left(q^{-x};a;q\right)\right]}{\nabla q^{-x}}=\frac{1}{1-q}w(x;aq;q)C_{n+1}\!\left(q^{-x};aq;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{a^{x}q^{\binomial{x+1}{2}}}{\left(q;q\right)_{x}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;a;q)C_{n}\!\left(q^{-x};a;q\right)=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;aq^{-n};q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\frac{a^{x}q^{\binomial{x+1}{2}}}{\left(q;q\right)_{x}}}}}$

${\displaystyle{\displaystyle{\displaystyle\nabla_{q}:=\frac{\nabla}{\nabla q^{-x}}}}}$

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}\,\qHyperrphis{1}{1}@@{q^{-x}}{0}{q}{-a^{-1}qt}=\sum_{n=0}^{\infty}\frac{C_{n}\!\left(q^{-x};a;q\right)}{\left(q;q\right)_{n}}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}\,\qHyperrphis{0}{1}@@{-}{-a^{-1}q}{q}{-a^{-1}q^{-x+1}t}=\sum_{n=0}^{\infty}\frac{C_{n}\!\left(q^{-x};a;q\right)}{\left(-a^{-1}q,q;q\right)_{n}}t^{n}}}}$

Limit relations

q-Meixner polynomial to q-Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(x;0,c;q\right)=C_{n}\!\left(x;c;q\right)}}}$

q-Krawtchouk polynomial to q-Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K_{n}\!\left(q^{-x};a^{-1}q^{-N},N;q\right)=C_{n}\!\left(q^{-x};a;q\right)}}}$

q-Charlier polynomial to Stieltjes-Wigert polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow\infty}C_{n}\!\left(ax;a;q\right)=\left(q;q\right)_{n}S_{n}\!\left(x;q\right)}}}$

q-Charlier polynomial to Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}C_{n}\!\left(q^{-x};a(1-q);q\right)=C_{n}\!\left(x;a\right)}}}$

Remark

${\displaystyle{\displaystyle{\displaystyle\frac{C_{n}\!\left(-x;-q^{-\alpha};q\right)}{\left(q;q\right)_{n}}=L^{(\alpha)}_{n}\!\left(x;q\right)}}}$

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q-Charlier

Al-Salam-Carlitz I >>