Discrete q-Hermite polynomials
# Discrete q-Hermite polynomials

In mathematics, the discrete q-Hermite polynomials are two closely related families $h_n(x;q)$ and $ĥ_n(x;q)$ of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam and Carlitz (1965). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. $h_n(x;q)$ is also called discrete q-Hermite I polynomials and $ĥ_n(x;q)$ is also called discrete q-Hermite II polynomials.

## Definition

The discrete q-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials by

    $\displaystyle h_{n}(x;q)=q^{\binom {n}{2}}_{2}\phi _{1}(q^{-n},x^{-1};0;q,-qx)=x^{n}_{2}\phi _{0}(q^{-n},q^{-n+1};;q^{2},q^{2n-1}/x^{2})=U_{n}^{(-1)}(x;q)$
    $\displaystyle {\hat {h}}_{n}(x;q)=i^{-n}q^{-{\binom {n}{2}}}_{2}\phi _{0}(q^{-n},ix;;q,-q^{n})=x^{n}_{2}\phi _{1}(q^{-n},q^{-n+1};0;q^{2},-q^{2}/x^{2})=i^{-n}V_{n}^{(-1)}(ix;q)$

and are related by

    $h_{n}(ix;q^{-1})=i^{n}{\hat {h}}_{n}(x;q)$
