Discrete q-Hermite polynomials

In mathematics, the discrete q-Hermite polynomials are two closely related families h_n(x;q) and ĥ_n(x;q) of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Al-Salam and Carlitz (1965). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition

The discrete q-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials by

$h_n(x;q)=q^{(\genfrac{}{}{0ex}{}{n}{2})}{}_2{\varphi }_1(q^{-n},x^{-1};0;q,-qx)=x^n{}_2{\varphi }_0(q^{-n},q^{-n+1};;q^2,q^{2n-1}/x^2)=U_n^{(-1)}(x;q)$

${\hat{h}}_n(x;q)=i^{-n}{q}^{-(\genfrac{}{}{0ex}{}{n}{2})}{}_2{\varphi }_0(q^{-n},ix;;q,-q^n)=x^n{}_2{\varphi }_1(q^{-n},q^{-n+1};0;q^2,-q^2/x^2)=i^{-n}{V}_n^{(-1)}(ix;q)$

and are related by

$h_n(ix;q^{-1})=i^n{\hat{h}}_n(x;q)$

References

• Berg, Christian; Ismael, Mourad (1994), Q-Hermite Polynomials and Classical Orthogonal Polynomials, arXiv:math/9405213
• Al-Salam, W. A.; Carlitz, L. (1965), "Some orthogonal q-polynomials", Mathematische Nachrichten, 30 (1–2): 47–61, doi:10.1002/mana.19650300105, ISSN 0025-584X, MR 0197804
• 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), "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