q-Laguerre polynomials

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In mathematics, the q-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials P(α)
n
(x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by Daniel S. Moak (1981). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.

Definition[edit]

The q-Laguerre polynomials are given in terms of basic hypergeometric functions and the Pochhammer symbol by

\displaystyle  L_n^{(\alpha)}(x;q) = \frac{(q^{\alpha+1};q)_n}{(q;q)_n} {}_1\phi_1(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)

Orthogonality[edit]

Recurrence and difference relations[edit]

Rodrigues formula[edit]

Generating function[edit]

Relation to other polynomials[edit]

References[edit]