On Certain Properties and Applications of the Perturbed Meixner–Pollaczek Weight

Abey S Kelil,Appanah R Appadu,Alta S Jooste

doi:10.3390/math9090955

Abey S Kelil, Appanah R Appadu + Show 1 more

Open Access

https://doi.org/10.3390/math9090955

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Journal: Mathematics	Publication Date: Apr 24, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: Nelson Mandela University, University of Pretoria

Abstract

This paper deals with monic orthogonal polynomials orthogonal with a perturbation of classical Meixner–Pollaczek measure. These polynomials, called Perturbed Meixner–Pollaczek polynomials, are described by their weight function emanating from an exponential deformation of the classical Meixner–Pollaczek measure. In this contribution, we investigate certain properties such as moments of finite order, some new recursive relations, concise formulations, differential-recurrence relations, integral representation and some properties of the zeros (quasi-orthogonality, monotonicity and convexity of the extreme zeros) of the corresponding perturbed polynomials. Some auxiliary results for Meixner–Pollaczek polynomials are revisited. Some applications such as Fisher’s information, Toda-type relations associated with these polynomials, Gauss–Meixner–Pollaczek quadrature as well as their role in quantum oscillators are also reproduced.

Highlights

Let us define some terminologies, notations and conventions that we will use throughout this paper
It is shown in Proposition 2 that the moments of the Meixner–Pollaczek measure are finite
We present a result proving the finiteness of moments of the perturbed

Summary

Introduction

Let us define some terminologies, notations and conventions that we will use throughout this paper. The set of complex numbers will be denoted by C and i will stand for the imaginary number (i2 = −1); the set of positive integers will be denoted by N, and N0 will denote the set of non-negative integers. All polynomials considered will be real-valued in one real variable, and P will stand for the set of all such polynomials. For each n ∈ N0 , the subset of P of all polynomials of degree not greater than n will be denoted by Pn. By a system of monic polynomials, we will mean a sequence {Φn }∞. N=0 of (n) polynomials satisfying Φn = n! A sequence of real polynomials {Φn }∞

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