Abstract |
In this contribution, we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product\n < f,g > s := < u,fg > + N(D(q)f)(alpha)(D(q)g)(alpha), alpha is an element of R, N >= 0,\n where u is a q-classical linear functional and D-q is the q-derivative operator. We obtain some algebraic properties of these polynomials such as an explicit representation, a five-term recurrence relation as well as a second order linear q-difference holonomic equation fulfilled by such polynomials. We present an analysis of the behaviour of its zeros as a function of the mass N. In particular, we obtain the exact values of N such that the smallest (respectively, the greatest) zero of the studied polynomials is located outside of the support of the measure. We conclude this work by considering two examples. |