# 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 Al-Salam and 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

$U_n^{(a)}(x;q) = (-a)^nq^{n(n-1)/2}{}_2\phi_1(q^{-n}, x^{-1};0;q,qx/a)$
$V_n^{(a)}(x;q) = (-a)^nq^{-n(n-1)/2}{}_2\phi_0(q^{-n}, x;;q,q^n/a)$

## References

• Wang, M. (2009). $q$-integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945.
• Askey, R., & Suslov, S. K. (1993). The $q$-harmonic oscillator and the Al-Salam and Carlitz polynomials. Letters in Mathematical Physics, 29(2), 123-132.
• Baker, T. H., & Forrester, P. J. (2000). Multivariable Al–Salam & Carlitz Polynomials Associated with the Type A $q$–Dunkl Kernel. Mathematische Nachrichten, 212(1), 5-35.category:Special functions