Al-Salam-Carlitz I

<< q-Charlier

Al-Salam-Carlitz I

Al-Salam-Carlitz II >>

Basic hypergeometric representation

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\int_{a}^{1}\left(qx,a^{-1}qx;q\right)_{\infty}U^{(a)}_{m}\!\left(x;q\right)U^{(a)}_{n}\!\left(x;q\right)\,d_{q}x{}=(-a)^{n}(1-q)\left(q;q\right)_{n}\left(q,a,a^{-1}q;q\right)_{\infty}q^{\binomial{n}{2}}\,\delta_{m,n}}}}$

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

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle xU^{(a)}_{n}\!\left(x;q\right)=U^{(a)}_{n+1}\!\left(x;q\right)+(a+1)q^{n}U^{(a)}_{n}\!\left(x;q\right)-aq^{n-1}(1-q^{n})U^{(a)}_{n-1}\!\left(x;q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{U}}^{(a)}_{n}\!\left(x\right)={\widehat{U}}^{(a)}_{n+1}\!\left(x\right)+(a+1)q^{n}{\widehat{U}}^{(a)}_{n}\!\left(x\right)-aq^{n-1}(1-q^{n}){\widehat{U}}^{(a)}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle U^{(a)}_{n}\!\left(x;q\right)={\widehat{U}}^{(a)}_{n}\!\left(x\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle(1-q^{n})x^{2}y(x)=aq^{n-1}y(qx)-\left[aq^{n-1}+q^{n}(1-x)(a-x)\right]y(x){}+q^{n}(1-x)(a-x)y(q^{-1}x)}}}$

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

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle U^{(a)}_{n}\!\left(x;q\right)-U^{(a)}_{n}\!\left(qx;q\right)=(1-q^{n})xU^{(a)}_{n-1}\!\left(x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}U^{(a)}_{n}\!\left(x;q\right)=\frac{1-q^{n}}{1-q}U^{(a)}_{n-1}\!\left(x;q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle aU^{(a)}_{n}\!\left(x;q\right)-(1-x)(a-x)U^{(a)}_{n}\!\left(q^{-1}x;q\right)=-q^{-n}xU^{(a)}_{n+1}\!\left(x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q^{-1}}\left[w(x;a;q)U^{(a)}_{n}\!\left(x;q\right)\right]=\frac{q^{-n+1}}{a(1-q)}w(x;a;q)U^{(a)}_{n+1}\!\left(x;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\left(qx,a^{-1}qx;q\right)_{\infty}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;a;q)U^{(a)}_{n}\!\left(x;q\right)=a^{n}q^{\frac{1}{2}n(n-3)}(1-q)^{n}\left(\mathcal{D}_{q^{-1}}\right)^{n}\left[w(x;a;q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;a;q)=\left(qx,a^{-1}qx;q\right)_{\infty}}}}$

Generating function

${\displaystyle{\displaystyle{\displaystyle\frac{\left(t,at;q\right)_{\infty}}{\left(xt;q\right)_{\infty}}=\sum_{n=0}^{\infty}\frac{U^{(a)}_{n}\!\left(x;q\right)}{\left(q;q\right)_{n}}t^{n}}}}$

Limit relations

Big q-Laguerre polynomial to Al-Salam-Carlitz I polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{a\rightarrow 0}\frac{P_{n}\!\left(aqx;a,ab;q\right)}{a^{n}}=q^{n}U^{(b)}_{n}\!\left(x;q\right)}}}$

Dual q-Krawtchouk polynomial to Al-Salam-Carlitz I polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K_{n}\!\left(\lambda(x);a^{-1},N|q\right)=\left(-\frac{1}{a}\right)^{n}q^{-\binomial{n}{2}}U^{(a)}_{n}\!\left(q^{x};q\right)}}}$

Al-Salam-Carlitz I polynomial to Discrete q-Hermite I polynomial

${\displaystyle{\displaystyle{\displaystyle U^{(-1)}_{n}\!\left(x;q\right)=h_{n}\!\left(x;q\right)}}}$

Al-Salam-Carlitz I polynomial to Charlier / Hermite polynomial

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

Remark

${\displaystyle{\displaystyle{\displaystyle U^{(a)}_{n}\!\left(x;q^{-1}\right)=V^{(a)}_{n}\!\left(x;q\right)}}}$

<< q-Charlier

Al-Salam-Carlitz I

Al-Salam-Carlitz II >>