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Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials. (English) Zbl 1405.33024
Summary: In this contribution, we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product \[ \langle f,g\rangle_s := \langle\mathbf{u},fg\rangle + N(\mathcal D_q f)(\alpha)(\mathcal D_q g)(\alpha),\quad \alpha\in\mathbb R,\, N\geq 0, \] where \(\mathbf{u}\) is a \(q\)-classical linear functional and \(\mathcal D_q\) 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.
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30 \(q\)-calculus and related topics
39A13 Difference equations, scaling (\(q\)-differences)
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