Discrete q-Hermite polynomials

In mathematics, the discrete q-Hermite polynomials are two closely related families h_n(x;q) and ĥ_n(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam and Carlitz (1965). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. h_n(x;q) is also called discrete q-Hermite I polynomials and ĥ_n(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

$h_n(x;q)=q^{(\genfrac{}{}{0ex}{}{n}{2})}{}_2{\varphi }_1(q^{-n},x^{-1};0;q,-qx)=x^n{}_2{\varphi }_0(q^{-n},q^{-n+1};;q^2,q^{2n-1}/x^2)=U_n^{(-1)}(x;q)$

${\hat{h}}_n(x;q)=i^{-n}{q}^{-(\genfrac{}{}{0ex}{}{n}{2})}{}_2{\varphi }_0(q^{-n},ix;;q,-q^n)=x^n{}_2{\varphi }_1(q^{-n},q^{-n+1};0;q^2,-q^2/x^2)=i^{-n}{V}_n^{(-1)}(ix;q)$

and are related by

$h_n(ix;q^{-1})=i^n{\hat{h}}_n(x;q)$

References

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