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 and the Pochhammer symbol.

Recurrence and difference relations

2xH_{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{aligned}H_{0}(x\mid q)&=1\\H_{1}(x\mid q)&=2x\\H_{2}(x\mid q)&=4x^{2}-(1-q^{n})\\H_{3}(x\mid q)&=8x^{3}-4x(1-q^{n})\\H_{4}(x\mid q)&=16x^{4}-12x^{2}(1-q^{n})+(1-q^{n})^{2}\\H_{5}(x\mid q)&=32x^{5}-32x^{3}(1-q^{n})+6x(1-q^{n})^{2}\end{aligned}}

Generating function

\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 $\textstyle x=\cos \theta$ .

References

Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, doi:10.2277/0521833574, 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), http://dlmf.nist.gov/18 |contribution-url= missing title (help), in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248

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