Continuous q-Hermite polynomials

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

$H_n(x|q)=e^{in\theta }{}_2{\varphi }_0[\begin{array}{c}{q}^{-n},0\\ -\end{array};q,q^n{e}^{-2i\theta }],x=\cos \theta .$

Recurrence and difference relations

$2x{H}_n(x\mid q)=H_{n+1}(x\mid q)+(1-q^n)H_{n-1}(x\mid q)$

with the initial conditions

$H_0(x\mid q)=1,H_{-1}(x\mid q)=0$

From the above, one can easily calculate:

$\begin{array}{cc}{H}_0(x\mid q)& =1\\ H_1(x\mid q)& =2x\\ H_2(x\mid q)& =4x^2-(1-q)\\ H_3(x\mid q)& =8x^3-2x(2-q-q^2)\\ H_4(x\mid q)& =16x^4-4x^2(3-q-q^2-q^3)+(1-q-q^3+q^4)\end{array}$

Generating function

$\sum _{n=0}^{\mathrm{\infty }}{H}_n(x\mid q)\frac{t^n}{(q;q{)}_n}=\frac{1}{{(t{e}^{i\theta },t{e}^{-i\theta };q)}_{\mathrm{\infty }}}$

where $x=\cos \theta$.

References

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