Al-Salam-Carlitz II

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Al-Salam-Carlitz II

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Basic hypergeometric representation

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

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{q^{k^{2}}a^{k}}{\left(q;q\right)_{k}\left(aq;q\right)_{k}}V^{(a)}_{m}\!\left(q^{-k};q\right)V^{(a)}_{n}\!\left(q^{-k};q\right){}=\frac{\left(q;q\right)_{n}a^{n}}{\left(aq;q\right)_{\infty}q^{n^{2}}}\,\delta_{m,n}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0<aq<1}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle xV^{(a)}_{n}\!\left(x;q\right)=V^{(a)}_{n+1}\!\left(x;q\right)+(a+1)q^{-n}V^{(a)}_{n}\!\left(x;q\right){}+aq^{-2n+1}(1-q^{n})V^{(a)}_{n-1}\!\left(x;q\right)}}}$

Monic recurrence relation

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

q-Difference equation

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

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

Forward shift operator

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

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle aV^{(a)}_{n}\!\left(x;q\right)-(1-x)(a-x)V^{(a)}_{n}\!\left(qx;q\right)=-q^{n}xV^{(a)}_{n+1}\!\left(x;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\mathcal{D}_{q}\left[w(x;a;q)V^{(a)}_{n}\!\left(x;q\right)\right]=-\frac{q^{n}}{a(1-q)}w(x;a;q)V^{(a)}_{n+1}\!\left(x;q\right)}}}$

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

Rodrigues-type formula

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

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

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{\left(xt;q\right)_{\infty}}{\left(t,at;q\right)_{\infty}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\binomial{n}{2}}}{\left(q;q\right)_{n}}V^{(a)}_{n}\!\left(x;q\right)t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\left(at;q\right)_{\infty}\cdot\qHyperrphis{1}{1}@@{x}{at}{q}{t}=\sum_{n=0}^{\infty}\frac{q^{n(n-1)}}{\left(q;q\right)_{n}}V^{(a)}_{n}\!\left(x;q\right)t^{n}}}}$

Limit relations

q-Meixner polynomial to Al-Salam-Carlitz II polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow 0}M_{n}\!\left(x;-ac^{-1},c;q\right)=\left(-\frac{1}{a}\right)^{n}q^{\binomial{n}{2}}V^{(a)}_{n}\!\left(x;q\right)}}}$

Quantum q-Krawtchouk polynomial to Al-Salam-Carlitz II polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}K^{\mathrm{qtm}}_{n}\!\left(x;a^{-1}q^{-N-1},N;q\right)=\left(-\frac{1}{a}\right)^{n}q^{\binomial{n}{2}}V^{(a)}_{n}\!\left(x;q\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle{\mathrm{i}^{-n}}V^{(-1)}_{n}\!\left(\mathrm{i}x;q\right)=\tilde{h}_{n}\!\left(x;q\right)}}}$

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

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

Remark

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

<< Al-Salam-Carlitz I

Al-Salam-Carlitz II

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