Al-Salam–Carlitz polynomials

In mathematics, Al-Salam–Carlitz polynomials U^(a)
_n(x;q) and V^(a)
_n(x;q) are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Waleed Al-Salam and Leonard Carlitz (1965). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14.24, 14.25) give a detailed list of their properties.

Definition

The Al-Salam–Carlitz polynomials are given in terms of basic hypergeometric functions by

${\displaystyle U_{n}^{(a)}(x;q)=(-a)^{n}q^{n(n-1)/2}{}_{2}\phi _{1}(q^{-n},x^{-1};0;q,qx/a)}$

${\displaystyle V_{n}^{(a)}(x;q)=(-a)^{n}q^{-n(n-1)/2}{}_{2}\phi _{0}(q^{-n},x;-;q,q^{n}/a)}$

References

• Al-Salam, W. A.; Carlitz, L. (1965), "Some orthogonal q-polynomials", Mathematische Nachrichten, 30: 47–61, doi:10.1002/mana.19650300105, ISSN 0025-584X, MR 0197804
• 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

Further reading

• Wang, M. (2009). ${\displaystyle q}$-integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945.
• Askey, R., & Suslov, S. K. (1993). The ${\displaystyle q}$-harmonic oscillator and the Al-Salam and Carlitz polynomials. Letters in Mathematical Physics, 29(2), 123-132.
• Chen, W. Y., Saad, H. L., & Sun, L. H. (2010). An operator approach to the Al-Salam–Carlitz polynomials. Journal of Mathematical Physics, 51(4).
• Kim, D. (1997). On combinatorics of Al-Salam Carlitz polynomials. European Journal of Combinatorics, 18(3), 295-302.
• Andrews, G. E. (2000). Schur's theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. Contemporary Mathematics, 254, 45-56.
• Baker, T. H., & Forrester, P. J. (2000). Multivariable Al–Salam & Carlitz Polynomials Associated with the Type A ${\displaystyle q}$–Dunkl Kernel. Mathematische Nachrichten, 212(1), 5-35.