In mathematics, the continuous "q"-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
Definition.
The polynomials are given in terms of basic hypergeometric functions by
formula_1
 formula_2
Recurrence and difference relations.
with the initial conditions
 formula_3
From the above, one can easily calculate:
 formula_4
 formula_5
Generating function.
where formula_6.