The zeros of polynomials orthogonal with respect to

$q$-integral on several intervals in the complex plane.

*(English)* Zbl 1061.33016
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 5th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 5–12, 2003. Sofia: Bulgarian Academy of Sciences (ISBN 954-84952-8-7/pbk). 178-188 (2004).

Summary: 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.

##### MSC:

33D45 | Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) |