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

Abstract

In this paper, polynomials which are orthogonal with respect to the inner product 〈p,r〉 S= ∑ k=0 ∞ p(q k)r(q k) (aq) k(aq;q) ∞ (q;q) k +λ ∑ k=0 ∞ (D qp)(q k)(D qr)(q k) (aq) k(aq;q) ∞ (q;q) k , where D q 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 { Q n ( x)/ p n ( x; a| q)} n on compact subsets of C⧹[0,1] is given, where Q n ( x) is the nth degree monic orthogonal polynomial with respect to the above inner product and p n ( x; a| q) denotes the monic little q-Laguerre polynomial of degree n.