Multiplier sequences, classes of generalized Bessel functions and open problems

TL;DR

This paper explores the zeros of generalized Bessel functions, develops new multiplier sequences, and presents integral representations, opening new research directions in the theory of multiplier sequences.

Contribution

It introduces a logarithmic multiplier sequence and provides integral representations of generalized Bessel functions, advancing the understanding of their zero distribution.

Findings

Development of a logarithmic multiplier sequence

Integral representations of generalized Bessel functions

Formulation of open problems and conjectures

Abstract

Motivated by the study of the distribution of zeros of generalized Bessel-type functions, the principal goal of this paper is to identify new research directions in the theory of multiplier sequences. The investigations focus on multiplier sequences interpolated by functions which are not entire and sums, averages and parametrized families of multiplier sequences. The main results include (i) the development of a `logarithmic' multiplier sequence and (ii) several integral representations of a generalized Bessel-type function utilizing some ideas of G. H. Hardy and L. V. Ostrovskii. The explorations and analysis, augmented throughout the paper by a plethora of examples, led to a number of conjectures and intriguing open problems.