Continuous q-Laguerre polynomials explained

In mathematics, the continuous q-Laguerre polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.

Definition

The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by ^[1] 。

P ^{(\alpha)} _{n} (x|q) = ((q ^{\alpha+1} ;q) _{n})/((q;q) _{n}) _{3} \phi _{2} (q ^{-n} ,q ^{\alpha/2+1/4} e ^{i\theta} ,q ^{\alpha/2+1/4} e ^{-i\theta} ;q ^{\alpha+1} , 0 |q,q)

Notes and References

1. Roelof Koekoek, Peter Lesky, Rene Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, p514, Springer