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Al-Salam–Carlitz polynomials

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Not to be confused with Al-Salam–Chihara polynomials.

In mathematics, Al-Salam–Carlitz polynomials U
n(x;q) and V
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

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

Further reading

  • Wang, M. (2009). q {\displaystyle q} -integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945.
  • Askey, R., & Suslov, S. K. (1993). The q {\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 q {\displaystyle q} –Dunkl Kernel. Mathematische Nachrichten, 212(1), 5-35.
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