The Zeros of Polynomials Orthogonal with Respect to $q$-Integral on Several Intervals in the Complex Plane

Rajkovic, Predrag M.; Stankovic, Miomir S.; Marinkovic, Sladjana D.

doi:10.7546/giq-5-2004-178-188

VOL. 5 | 2004 The Zeros of Polynomials Orthogonal with Respect to $q$-Integral on Several Intervals in the Complex Plane

Predrag M. Rajkovic, Miomir S. Stankovic, Sladjana D. Marinkovic

Editor(s) Ivaïlo M. Mladenov, Allen C. Hirshfeld

Geom. Integrability & Quantization, 2004: 178-188 (2004) DOI: 10.7546/giq-5-2004-178-188

Abstract

We construct the sequence of orthogonal polynomials with respect to an inner product defined in the sense of $q$-integration over several intervals in the complex plane. For such introduced polynomials we prove that all zeros lie in the smallest convex hull over the intervals in the complex plane. The results are stated precisely in some special cases, as some symmetric cases of equal lengths, angles and weights.