q-SOBOLEV ORTHOGONALITY OF THE q-LAGUERRE POLYNOMIALS {Ln(-N)(.; q)}n=0∞ FOR POSITIVE INTEGERS N

The family of q-Laguerre polynomials {L-n((alpha))(.; q)}(n=0)(infinity)is usually defined for 0 < q < 1 and alpha > -1. We extend this family to a new one in which arbitrary complex values of the parameter a are allowed. These so-called generalized q-Laguerre polynomials fulfil the same three term recurrence relation as the original ones, but when the parameter a 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))(.; q)}(n=0)(infinity), for positive integers N, become orthogonal.