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

${\displaystyle H_{n}(x|q)=e^{in\theta }{}_{2}\phi _{0}\left[{\begin{matrix}q^{-n},0\\-\end{matrix}};q,q^{n}e^{-2i\theta }\right],\quad x=\cos \,\theta .}$

Recurrence and difference relations

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

with the initial conditions

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

From the above, one can easily calculate:

${\displaystyle {\begin{aligned}H_{0}(x\mid q)&=1\\H_{1}(x\mid q)&=2x\\H_{2}(x\mid q)&=4x^{2}-(1-q)\\H_{3}(x\mid q)&=8x^{3}-2x(2-q-q^{2})\\H_{4}(x\mid q)&=16x^{4}-4x^{2}(3-q-q^{2}-q^{3})+(1-q-q^{3}+q^{4})\end{aligned}}}$

Generating function

${\displaystyle \sum _{n=0}^{\infty }H_{n}(x\mid q){\frac {t^{n}}{(q;q)_{n}}}={\frac {1}{\left(te^{i\theta },te^{-i\theta };q\right)_{\infty }}}}$

where ${\displaystyle \textstyle x=\cos \theta }$.

References

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
• Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
• Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Chapter 18: Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
• Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. CiteSeerX 10.1.1.643.3896.