Reproducing kernel Hilbert spaces generated by the binomial coefficients

TL;DR

This paper explores a specific reproducing kernel Hilbert space linked to binomial coefficients, introducing transforms for harmonic analysis and discussing connections to discrete analytic functions and quantum theory.

Contribution

It develops new transforms enabling harmonic analysis in the RKHS generated by binomial coefficients, and links this space to discrete analytic functions and quantum theory.

Findings

Introduction of two transforms for harmonic analysis

Establishment of connections with discrete analytic functions

Discussion of links to quantum theory

Abstract

We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space. Finally, we mention connections with the theory of discrete analytic functions and with the quantum case.