Lima, Helder, Loureiro, Ana F. (2020) Multiple orthogonal polynomials with respect to Gauss' hypergeometric function. arXiv, . (Submitted) (KAR id:85292)
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Abstract
A new set of multiple orthogonal polynomials of both type I and type II with respect to two weight functions involving Gauss' hypergeometric function on the interval (0, 1) is studied. This type of polynomials have direct applications in the investigation of singular values of products of Ginibre matrices, in the analysis of rational solutions to Painlevé equations and are connected with branched continued fractions and total positivity problems in combinatorics. The pair of orthogonality measures is shown to be a Nikishin system and to satisfy a matrix Pearsontype differential equation. The focus is on the polynomials whose indexes lie on the step line, for which it is shown that differentiation on the variable gives a shift on the parameters, therefore satisfying Hahn's property. We obtain a Rodriguestype formula for type I, while a more detailed characterisation is given for the type II polynomials (aka 2orthogonal polynomials) which include: an explicit expression as a terminating hypergeometric series, a thirdorder differential equation, and a thirdorder recurrence relation. The asymptotic behaviour of their recurrence coefficients mimics those of JacobiPiñeiro polynomials, based on which, their zero asymptotic distribution and a MehlerHeine asymptotic formula near the origin are given. Particular choices on the parameters degenerate in some known systems such as special cases of the JacobiPiñeiro polynomials, Jacobitype 2orthogonal polynomials, and components of the cubic decomposition of threefold symmetric Hahnclassical polynomials. Equally considered are confluence relations to other known polynomial sets, such as multiple orthogonal polynomials with respect to Tricomi functions.
Item Type:  Article 

Uncontrolled keywords:  Multiple orthogonal polynomials, Gauss hypergeometric function, Nikishin system, Rodriguestype formula, generalised hypergeometric series, 2orthogonal polynomials, Hahn classical 
Subjects: 
Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Q Science > QA Mathematics (inc Computing science) > QA351 Special functions 
Divisions:  Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science 
Depositing User:  Ana F. Loureiro 
Date Deposited:  04 Jan 2021 01:06 UTC 
Last Modified:  16 Feb 2021 14:17 UTC 
Resource URI:  https://kar.kent.ac.uk/id/eprint/85292 (The current URI for this page, for reference purposes) 
Loureiro, Ana F.:  https://orcid.org/0000000241378822 
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