In mathematics, the discrete q-Hermite polynomials are two closely related families hn(x;q) and ĥn(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. hn(x;q) is also called discrete q-Hermite I polynomials and ĥn(x;q) is also called discrete q-Hermite II polynomials.
Definition[edit]
The discrete q-Hermite polynomials are given in terms of basic hypergeometric functions and the Al-Salam–Carlitz polynomials by
and are related by
References[edit]
- 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, vol. 96 (2nd ed.), Cambridge University Press, ISBN 978-0-521-83357-8, MR 2128719
- Jazmati, M. Saleh; Mezlini, Kamel; Bettaibi, Neji (2014), "Generalized q-Hermite polynomials and the q-Dunkl heat equation", Bulletin of Mathematical Analysis and Applications, 6 (4), Prishtine, Serbia: Prishtine: Department of Mathematics and Computer Sciences: 16–43, ISSN 1821-1291
- 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.