See also: big q-Laguerre polynomials, continuous q-Laguerre polynomials and little q-Laguerre polynomials. In mathematics, the q-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials P(x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by . give a detailed list of their properties.
The q-Laguerre polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by
\displaystyle
| (\alpha) | |
| L | |
| n |
(x;q)=
| (q\alpha+1;q)n | |
| (q;q)n |
{}1\phi
| -n | |
| 1(q |
;q\alpha+1;q,-qn+\alpha+1x).
Orthogonality is defined by the unimono nature of the polynomials' convergence at boundaries in integral form.