Continuous big q-Hermite polynomials explained

In mathematics, the continuous big q-Hermite polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions.

H ^{- n} _{n(x;a|q) = a} {} _{3} \phi _{2\left[\begin{matrix}} q ^{-n} ,ae ^{i\theta} ,ae ^{-i\theta} \ 0 , 0 \end{matrix} ;q,q\right], x = \cos\theta.