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

[math]\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. }[/math]

Recurrence and difference relations

[math]\displaystyle{ 2x H_n(x\mid q) = H_{n+1} (x\mid q) + (1-q^n) H_{n-1} (x\mid q) }[/math]

with the initial conditions

[math]\displaystyle{ H_0 (x\mid q) =1, H_{-1} (x\mid q) = 0 }[/math]

From the above, one can easily calculate:

[math]\displaystyle{ \begin{align} 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{align} }[/math]

Generating function

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

where [math]\displaystyle{ \textstyle x=\cos \theta }[/math].

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.