Continuous q-Hermite polynomials

Summary

The continuous $q$-Hermite polynomials $H_n(x \mid q)$ are a family of orthogonal polynomials on $[-1, 1]$ depending on a base $q$. They are $q$-analogues of the Hermite polynomials, to which they tend after rescaling as $q \to 1$. At $q = 0$ they are the Chebyshev polynomials of the second kind. They are one of the simplest families of the $q$-Askey scheme, a special case of the Rogers polynomials and of the Askey–Wilson polynomials. They appear in $q$-deformed probability and in the theory of $q$-Gaussian processes.