Continuous big q-Hermite polynomials

Short description: A family of basic hypergeometric orthogonal polynomials

In mathematics, the continuous big 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. [math]\displaystyle{ H_n(x;a|q)=a^{-n}{}_{3}\phi_2\left[\begin{matrix} q^{-n},ae^{i\theta},ae^{-i\theta}\\ 0,0\end{matrix} ;q,q\right],\quad x=\cos\,\theta. }[/math]

References

• Floreanini, Roberto; LeTourneux, Jean; Vinet, Luc (1995), "An algebraic interpretation of the continuous big q-Hermite polynomials", Journal of Mathematical Physics (AIP Publishing) 36 (9): 5091–5097, doi:10.1063/1.531216, ISSN 1089-7658, Bibcode: 1995JMP....36.5091F 
• 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