Q-Meixner

<< Little q-Jacobi: Special case

q-Meixner

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Basic hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x};b,c;q\right)=\qHyperrphis{2}{1}@@{q^{-n},q^{-x}}{bq}{q}{-\frac{q^{n+1}}{c}}}}}$

Orthogonality relation(s)

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

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle 0\leq bq<1}}}$ &
${\displaystyle{\displaystyle{\displaystyle c>0}}}$

Recurrence relation

${\displaystyle{\displaystyle{\displaystyle q^{2n+1}(1-q^{-x})M_{n}\!\left(q^{-x}\right){}=c(1-bq^{n+1})M_{n+1}\!\left(q^{-x}\right){}-\left[c(1-bq^{n+1})+q(1-q^{n})(c+q^{n})\right]M_{n}\!\left(q^{-x}\right){}+q(1-q^{n})(c+q^{n})M_{n-1}\!\left(q^{-x}\right)}}}$
${\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x}\right):=M_{n}\!\left(q^{-x};b,c;q\right)}}}$

Monic recurrence relation

${\displaystyle{\displaystyle{\displaystyle x{\widehat{M}}_{n}\!\left(x\right)={\widehat{M}}_{n+1}\!\left(x\right)+\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^{n})(c+q^{n})\right\}\right]{\widehat{M}}_{n}\!\left(x\right){}+cq^{-4n+1}(1-q^{n})(1-bq^{n})(c+q^{n}){\widehat{M}}_{n-1}\!\left(x\right)}}}$
${\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x};b,c;q\right)=\frac{(-1)^{n}q^{n^{2}}}{\left(bq;q\right)_{n}c^{n}}{\widehat{M}}_{n}\!\left(q^{-x}\right)}}}$

q-Difference equation

${\displaystyle{\displaystyle{\displaystyle-(1-q^{n})y(x)=B(x)y(x+1)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-1)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle D(x)=(1-q^{x})(1+bcq^{x})}}}$ &

${\displaystyle{\displaystyle{\displaystyle B(x)=cq^{x}(1-bq^{x+1})}}}$ &

${\displaystyle{\displaystyle{\displaystyle y(x)=M_{n}\!\left(q^{-x};b,c;q\right)}}}$

Forward shift operator

${\displaystyle{\displaystyle{\displaystyle M_{n}\!\left(q^{-x-1};b,c;q\right)-M_{n}\!\left(q^{-x};b,c;q\right){}=-\frac{q^{-x}(1-q^{n})}{c(1-bq)}M_{n-1}\!\left(q^{-x};bq,cq^{-1};q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\Delta M_{n}\!\left(q^{-x};b,c;q\right)}{\Delta q^{-x}}=-\frac{q(1-q^{n})}{c(1-q)(1-bq)}M_{n-1}\!\left(q^{-x};bq,cq^{-1};q\right)}}}$

Backward shift operator

${\displaystyle{\displaystyle{\displaystyle cq^{x}(1-bq^{x})M_{n}\!\left(q^{-x};b,c;q\right)-(1-q^{x})(1+bcq^{x})M_{n}\!\left(q^{-x+1};b,c;q\right){}=cq^{x}(1-b)M_{n+1}\!\left(q^{-x};bq^{-1},cq;q\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{\nabla\left[w(x;b,c;q)M_{n}\!\left(q^{-x};b,c;q\right)\right]}{\nabla q^{-x}}{}=\frac{1}{1-q}w(x;bq^{-1},cq;q)M_{n+1}\!\left(q^{-x};bq^{-1},cq;q\right)}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;b,c;q)=\frac{\left(bq;q\right)_{x}}{\left(q,-bcq;q\right)_{x}}c^{x}q^{\binomial{x+1}{2}}}}}$

Rodrigues-type formula

${\displaystyle{\displaystyle{\displaystyle w(x;b,c;q)M_{n}\!\left(q^{-x};b,c;q\right)=(1-q)^{n}\left(\nabla_{q}\right)^{n}\left[w(x;bq^{n},cq^{-n};q)\right]}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle w(x;b,c;q)=\frac{\left(bq;q\right)_{x}}{\left(q,-bcq;q\right)_{x}}c^{x}q^{\binomial{x+1}{2}}}}}$

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

Generating functions

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}\,\qHyperrphis{1}{1}@@{q^{-x}}{bq}{q}{-c^{-1}qt}=\sum_{n=0}^{\infty}\frac{M_{n}\!\left(q^{-x};b,c;q\right)}{\left(q;q\right)_{n}}t^{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1}{\left(t;q\right)_{\infty}}\,\qHyperrphis{1}{1}@@{-b^{-1}c^{-1}q^{-x}}{-c^{-1}q}{q}{bqt}{}=\sum_{n=0}^{\infty}\frac{\left(bq;q\right)_{n}}{\left(-c^{-1}q,q;q\right)_{n}}M_{n}\!\left(q^{-x};b,c;q\right)t^{n}}}}$

Limit relations

Big q-Jacobi polynomial to q-Meixner polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow-\infty}\bigqJacobi{n}@{q^{-x}}{a}{-a^{-1}cd^{-1}}{c}{q}=\qMeixner{n}@{q^{-x}}{a}{d}{q}}}}$

q-Hahn polynomial to q-Meixner polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{N\rightarrow\infty}Q_{n}(q^{-x};b,-b^{-1}c^{-1}q^{-N-1},N|q)=M_{n}\!\left(q^{-x};b,c;q\right)}}}$

q-Meixner polynomial to q-Laguerre polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{c\rightarrow\infty}M_{n}\!\left(cq^{\alpha}x;q^{\alpha},c;q\right)=\frac{\left(q;q\right)_{n}}{\left(q^{\alpha+1};q\right)_{n}}L^{(\alpha)}_{n}\!\left(x;q\right)}}}$

q-Meixner polynomial to q-Charlier polynomial

${\displaystyle{\displaystyle{\displaystyle\qMeixner{n}@{x}{0}{c}{q}=\qCharlier{n}@{x}{c}{q}}}}$

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

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

q-Meixner polynomial to Meixner polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{q\rightarrow 1}M_{n}\!\left(q^{-x};q^{\beta-1},(1-c\right)^{-1}c;q)=M_{n}\!\left(x;\beta,c\right)}}}$

Remarks

${\displaystyle{\displaystyle{\displaystyle\qMeixner{n}@{q^{-x}}{b}{c}{q}=\littleqJacobi{n}@{-c^{-1}q^{n}}{b}{b^{-1}q^{-n-x-1}}{q}}}}$
${\displaystyle{\displaystyle{\displaystyle\qtmqKrawtchouk{n}@{q^{-x}}{p}{N}{q}=\qMeixner{n}@{q^{-x}}{q^{-N-1}}{-p^{-1}}{q}}}}$

<< Little q-Jacobi: Special case

q-Meixner

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