CAOP

q-Meixner Polynomials

Definition

The q-Meixner polynomials are defined as

\[ \begin{align} M_n(q^{-x};b,c;q) &= \sum_{k=0}^\infty \frac{(q^{-n};q)_k (q^{-x};q)_k}{(bq;q)_k (q;q)_k} \left(-\frac{q^{n+1}}{c}\right)^k\\ &= {}_{2}\phi_{1}\!\left(\left. {q^{-n}, q^{-x} \atop bq} \; \right| q ; -\frac{q^{n+1}}{c} \right) \end{align} \]

q-Difference Equation

q-Recurrence Equation

Parameters

Variables

\(n\)

\(x\)

\(q\)

Parameters

\(b\) \(0 < b < q^{-1}\)

\(c\) \(c > 0\)

factor (use Maple-style input)

   q-hypergeometric term in \(n\) and q-hypergeometric term in \(q^{-x}\) required