On the generating functions and special functions associated with superoscillations

TL;DR

This paper explores generating functions for superoscillatory coefficients, revealing new links to special polynomials and functions, and deriving recurrence and derivative formulas to deepen understanding of superoscillations in weak measurements.

Contribution

It introduces novel generating functions and relations connecting superoscillatory coefficients with well-known special functions, along with recurrence and derivative formulas.

Findings

Established new relations with Hermite polynomials and Stirling numbers

Developed recurrence relations for superoscillatory coefficients

Derived derivative formulas using generating functions

Abstract

The aim of this paper is to study generating functions for the coefficients of the classical superoscillatory function associated with weak measurements. We also establish some new relations between the superoscillatory coefficients and many well-known families of special polynomials, numbers, and functions such as Bernstein basis functions, the Hermite polynomials, the Stirling numbers of second kind, and also the confluent hypergeometric functions. Moreover, by using generating functions, we are able to develop a recurrence relation and a derivative formula for the superoscillatory coefficients.