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. h_n(x;q) is also called discrete q-Hermite I polynomials and ĥ_n(x;q) is also called discrete q-Hermite II polynomials.

Definition

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

[math]\displaystyle{ \displaystyle h_n(x;q)=q^{\binom{n}{2}}{}_2\phi_1(q^{-n},x^{-1};0;q,-qx) = x^n{}_2\phi_0(q^{-n},q^{-n+1};;q^2,q^{2n-1}/x^2) = U_n^{(-1)}(x;q) }[/math]

[math]\displaystyle{ \displaystyle \hat h_n(x;q)=i^{-n}q^{-\binom{n}{2}}{}_2\phi_0(q^{-n},ix;;q,-q^n) = x^n{}_2\phi_1(q^{-n},q^{-n+1};0;q^2,-q^{2}/x^2) = i^{-n}V_n^{(-1)}(ix;q) }[/math]

and are related by

[math]\displaystyle{ h_n(ix;q^{-1}) = i^n\hat h_n(x;q) }[/math]

References

• Berg, Christian; Ismael, Mourad (1994), Q-Hermite Polynomials and Classical Orthogonal Polynomials 
• 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 
• 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 
• Jazmati, M. Saleh; Mezlini, Kamel; Bettaibi, Neji (2014), "Generalized q-Hermite polynomials and the q-Dunkl heat equation", Bulletin of Mathematical Analysis and Applications (Prishtine, Serbia: Prishtine: Department of Mathematics and Computer Sciences) 6 (4): 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 
• 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