Analytic properties of some basic hypergeometric-Sobolev-type orthogonal polynomials

2018/08/31 by Roberto S. Costas-Santos, Costas-Santos, Roberto S., Anier Soria-Lorente +1
Mathematics · Social Sciences · #Mathematical functions and polynomials #Diverse Research Studies Overview #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.1809.08973

Abstract

In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product ⟨ f,g ⟩ _S:= ⟨ \bf u, f g⟩ +N (\mathscr D_q f)(α) (\mathscr D _qg)(α), α∈ \mathbb R, N≥ 0, where \bf u is a q-classical linear functional and \mathscr 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 function of the mass N. In particular, we in 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 considering two examples.

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