Journal of the
Korean Mathematical Society JKMS

The family of $q$-Laguerre polynomials $\{L_n^{(\alpha)}(\cdot;q)\}_{n=0}^{\infty}$ is usually defined for $0 < q-1$. We extend this family to a new one in which arbitrary complex values of the parameter $\alpha$ are allowed. These so-called generalized $q$-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter $\alpha$ is a negative integer, no orthogonality property can be deduced from Favard's theorem. In this work we introduce non-standard inner products involving $q$-derivatives with respect to which the generalized $q$-Laguerre polynomials $\{L_n^{(-N)}(\cdot;q)\}_{n=0}^{\infty}$, for positive integers $N$, become orthogonal.

Keywords: non-standard orthogonality, $q$-Laguerre polynomials, basic hypergeometric series

MSC numbers: 33D45, 42C05