Al-Salam–Carlitz polynomials

Not to be confused with Al-Salam–Chihara 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

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