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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.
##### MSC:
 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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