Generalizations of a q-analogue of Laguerre polynomials

AbstractWe study polynomials {Lnα, M0, M1,…, MN(x; q)}n = 0∞ orthogonal with respect to the inner product <f, g> = Гq(−α)Г(−α) Г(α + 1) ∫∞0xα(−(1−q)x;q)∞f(x) g(x) dx + ∑v = 0N Mv(Dvqg)(0), where α > − 1, N is an integer, and Mv ⩾ 0 for all v ϵ {0, 1, 2, …, N}. These polynomials are q-analogues of the polynomials {Lnα, M0, M1,…, MN(x)}n = 0∞ orthogonal with respect to the (Sobolev) inner product <f, g> = 1Г(α + 1)∫∞0 xαe−xf(x) g(x) dx + ∑v = 0N Mvf(v)(0) g(v)(0). We prove the orthogonality relation for which we give a discrete form (q-integral) too. We give a representation as a basic hypergeometric series, a recurrence relation is derived, a Christoffel-Darboux type formula and a second order q-difference equation satisfied by these new basic orthogonal polynomials