CAOP

q-Meixner Pollaczek Polynomials

Definition

The q-Meixner Pollaczek polynomials are defined as

\[ \begin{align} P_n(x;a|q) &= a^{-n} e^{-inu} \frac{(a^2;q)_n}{(q;q)_n} \sum_{k=0}^\infty \frac{(q^{-n};q)_k (a e^{i(t+2u)};q)_k (a e^{-it};q)_k}{(a^2;q)_k (q;q)_k} q^k\\ &= a^{-n} e^{-inu} \frac{(a^2;q)_n}{(q;q)_n} {}_{3}\phi_{2}\!\left(\left. {q^{-n}, a e^{i(t+2u)}, a e^{-it} \atop a^2, 0} \; \right| q ; q \right),\quad x=\cos(t+u) \end{align} \]

q-Recurrence Equation

Parameters

Variables

\(n\)

\(x\)

\(q\)

Parameters

\(a\) \(\)

factor (use Maple-style input)

   q-hypergeometric term in \(n\)