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 $$\multline\langle p,r\rangle_S=\sum infty_{k=0}p(q )r(q ) {(aq) (aq;q)_\infty\over(q;q)_k}\\ +\lambda\sum infty_{k=0} (D_qp)(q )(D_qr)(q ){(aq) (aq;q)_\infty\over(q;q)_k},\endmultline$$ where $D_q$ is the $q$-difference operator, $\lambda\geq0,\ 0<q<1$ 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 of the zeros of these polynomials are also deduced. Finally, the relative asymptotics $\{Q_n(x)/p_n(x;a )\}_n$ on compact subsets of ${\bf C}\sbs[0,1]$ is given, where $Q_n(x)$ is the $n$th degree monic orthogonal polynomial with respect to the above inner product and $p_n(x;a )$ denotes the monic little $q$-Laguerre polynomial of degree $n$.