In mathematics, the discrete q-Hermite polynomials are two closely related families hn(x;q) and ĥn(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. hn(x;q) is also called discrete q-Hermite I polynomials and ĥn(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
and are related by
References