Inner products involving q-differences: the little q-Laguerre–Sobolev polynomials

In this paper, polynomials which are orthogonal with respect to the inner product〈p,r〉S=∑k=0∞ p(qk)r(qk)(aq)k(aq;q)∞(q;q)k+λ∑k=0∞ (Dqp)(qk)(Dqr)(qk)(aq)k(aq;q)∞(q;q)k,where Dq is the q-difference operator, λ⩾0, 0<q<1 and 0<aq<1 are studied. For these polynomials, algebraic properties and q-difference equations are obtained as well as their relation with the monic little q-Laguerre polynomials. Some properties about the zeros of these polynomials are also deduced. Finally, the relative asymptotics {Qn(x)/pn(x;a|q)}n on compact subsets of C⧹[0,1] is given, where Qn(x) is the nth degree monic orthogonal polynomial with respect to the above inner product and pn(x;a|q) denotes the monic little q-Laguerre polynomial of degree n.