In mathematics, the discrete q-Hermite polynomials are two closely related families h<sub>n</sub>(x;q) and ÃÂ¥<sub>n</sub>(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. h<sub>n</sub>(x;q) is also called discrete q-Hermite I polynomials and ÃÂ¥<sub>n</sub>(x;q) is also called discrete q-Hermite II polynomials.
The discrete q-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-SalamâÂÂCarlitz polynomials by
and are related by