Inner products involving \(q\)-differences: The little \(q\)-Laguerre-Sobolev polynomials
Polynomials are considered that are orthogonal with respect to the inner product \(\langle p,r\rangle_S=\sum_{k=0}^\infty p(q^k)r(q^k){(aq)^k(aq;q)_\infty\over (q;q)_k }+\lambda\sum_{k=0}^\infty (D_qp)(q^k)(D_qr)(q^k) {(aq)^k(aq;q)_\infty\over (q;q)_k }\), where \(D_q\) is the \(q\)-difference operator, \(\lambda\geq 0\), \(0<q<1\) and \(0<aq<1\). For these polyonomials \(q\)-difference equations are obtained and their relation with the monic \(q\)-Laguerre polynomials. Some properties of the zeros of the polynomials and some asymptotic properties are given.
scientific article; zbMATH DE number 1472328