q-Meixner
Basic hypergeometric representation
Orthogonality relation(s)
Recurrence relation
![{\displaystyle {\displaystyle
q^{2n+1}(1-q^{-x})\qMeixner{n}@@{q^{-x}}{b}{c}{q}
{}=c(1-bq^{n+1})\qMeixner{n+1}@@{q^{-x}}{b}{c}{q}
{}-\left[c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right]\qMeixner{n}@@{q^{-x}}{b}{c}{q}
{}+q(1-q^n)(c+q^n)\qMeixner{n-1}@@{q^{-x}}{b}{c}{q}
}}](/index.php?title=Special:MathShowImage&hash=9974ee4cbd745f4b707c5e494f42df0e&mode=latexml)
Monic recurrence relation
![{\displaystyle {\displaystyle
x\monicqMeixner{n}@@{x}{b}{c}{q}=\monicqMeixner{n+1}@@{x}{b}{c}{q}+
\left[1+q^{-2n-1}\left\{c(1-bq^{n+1})+q(1-q^n)(c+q^n)\right\}\right]\monicqMeixner{n}@@{x}{b}{c}{q}
{}+cq^{-4n+1}(1-q^n)(1-bq^n)(c+q^n)\monicqMeixner{n-1}@@{x}{b}{c}{q}
}}](/index.php?title=Special:MathShowImage&hash=6ed537b1f8976604b865c50c4610465c&mode=latexml)
q-Difference equation
Forward shift operator

Backward shift operator

Rodrigues-type formula
Generating functions

Limit relations
Big q-Jacobi polynomial to q-Meixner polynomial

q-Hahn polynomial to q-Meixner polynomial

q-Meixner polynomial to q-Laguerre polynomial
![{\displaystyle {\displaystyle
\lim_{c\rightarrow\infty}\qMeixner{n}@{cq^{\alpha}x}{q^{\alpha}}{c}{q}=
\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q^{\alpha+1}}{q}{n}}\qLaguerre[\alpha]{n}@{x}{q}
}}](/index.php?title=Special:MathShowImage&hash=a8210e50a575324f46ea630cd30cc77a&mode=latexml)
q-Meixner polynomial to q-Charlier polynomial

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

q-Meixner polynomial to Meixner polynomial
