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 by

$\displaystyle$

Recurrence and difference relations

$2x H_n(x|q) = H_{n+1} (x|q) + (1-q^n) H_{n-1} (x|q)$ with the initial conditions $\textstyle H_{0} (x|q) =1, H_{-1} (x|q) = 0$

From the above, one can easily calculate:

$H_{0} (x|q) =1$

$H_{1} (x|q) = 2x$

$H_{2} (x|q) =4x^2 - (1-q^n)$

$H_{3} (x|q) =8x^3 - 4x(1-q^n)$

$H_{4} (x|q) =16x^4 - 12x^2(1-q^n) + (1-q^n)^2$

$H_{5} (x|q) =32x^5 - 32x^3(1-q^n) +6x(1-q^n)^2$

Generating function

$\displaystyle \sum_{n=0}^{\infty} H_n(x |q) \frac{t^n}{(q;q)_n} = \frac{1} {\left( t e^{i \theta},t e^{-i \theta};q \right)_{\infty}}$

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