Continuous big q-Hermite 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.

H

n

(

x

;

a

|

q

)

=

a

−

n

3

ϕ

2

[

q

−

n

,

a

e

i

θ

,

a

e

−

i

θ

0

,

0

;

q

,

q

]

,

x

=

cos

θ

.

{\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 .}

References

Floreanini, Roberto; LeTourneux, Jean; Vinet, Luc (1995), "An algebraic interpretation of the continuous big q-Hermite polynomials", Journal of Mathematical Physics, 36 (9), AIP Publishing: 5091–5097, arXiv:math/9504217, Bibcode:1995JMP....36.5091F, doi:10.1063/1.531216, ISSN 1089-7658, S2CID 15208438

Gasper, George; Rahman, Mizan (2004), Basic hypergeometric series, Encyclopedia of Mathematics and its Applications, vol. 96 (2nd ed.), Cambridge University Press, 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), "Chapter 18: Orthogonal Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.