# Multiple orthogonal polynomials associated with confluent hypergeometric functions

Lima, Hélder, Loureiro, Ana (2020) Multiple orthogonal polynomials associated with confluent hypergeometric functions. Journal of Approximation Theory, 260 . Article Number 105484. ISSN 0021-9045. (doi:10.1016/j.jat.2020.105484) (KAR id:82879)

## Abstract

We introduce and analyse a new family of multiple orthogonal polynomials of hypergeometric type with respect to two measures supported on the positive real line which can be described in terms

of confluent hypergeometric functions of the second kind. These two measures form a Nikishin system. Our focus is on the multiple orthogonal polynomials for indices on the step line. The sequences of the derivatives of both type I and type II polynomials with respect to these indices are again multiple orthogonal and they correspond to the original sequences with shifted parameters. For the type I polynomials, we provide a Rodrigues-type formula. We characterise the type II polynomials on the step line, also known as d-orthogonal polynomials (where d is the number of measures involved so that here d = 2), via their explicit expression as a terminating generalised hypergeometric series, as solutions to a third-order differential equation and via their recurrence relation. The latter involves recurrence coefficients which are unbounded and asymptotically periodic. Based on this information we deduce the asymptotic behaviour of the largest zeros of the type II polynomials. We also discuss limiting relations between these polynomials and the multiple orthogonal polynomials with respect to the modified Bessel weights. Particular choices on the parameters

for the 2-orthogonal polynomials under discussion correspond to the cubic components of the already known threefold symmetric Hahn-classical multiple orthogonal polynomials on star-like sets.

Item Type: Article 10.1016/j.jat.2020.105484 Multiple orthogonal polynomials, confluent hypergeometric function, Nikishin system, Rodrigues formula, generalised hypergeometric series, 2-orthogonal polynomials, Hahn-classical, 3-fold symmetric Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, CalculusQ Science > QA Mathematics (inc Computing science) > QA351 Special functions Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Ana F. Loureiro 12 Sep 2020 15:34 UTC 21 Sep 2021 23:00 UTC https://kar.kent.ac.uk/id/eprint/82879 (The current URI for this page, for reference purposes) https://orcid.org/0000-0002-4137-8822