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:

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

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

• Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8 
• 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 
• 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. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, http://dlmf.nist.gov/18 
• Sadjang, Patrick Njionou. Moments of Classical Orthogonal Polynomials (Ph.D.). Universität Kassel. CiteSeerX 10.1.1.643.3896.