Al-Salam–Carlitz polynomials
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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
- [math]\displaystyle{ U_n^{(a)}(x;q) = (-a)^nq^{n(n-1)/2}{}_2\phi_1(q^{-n}, x^{-1};0;q,qx/a) }[/math]
- [math]\displaystyle{ V_n^{(a)}(x;q) = (-a)^nq^{-n(n-1)/2}{}_2\phi_0(q^{-n}, x;-;q,q^n/a) }[/math]
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
- 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
Further reading
- Wang, M. (2009). [math]\displaystyle{ q }[/math]-integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945.
- Askey, R., & Suslov, S. K. (1993). The [math]\displaystyle{ q }[/math]-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 [math]\displaystyle{ q }[/math]–Dunkl Kernel. Mathematische Nachrichten, 212(1), 5-35.
category:Special functions
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