A Generalization of the Binomial Interpolated Operator and its Action on Linear Recurrent Sequences

TL;DR

This paper explores a generalized binomial interpolated operator's effects on linear recurrent sequences, revealing how it alters characteristic polynomial zeros, forms a group, and relates to well-known integer sequences like Fibonacci and Catalan numbers.

Contribution

It introduces a generalized operator acting on linear recurrent sequences, analyzes its properties, and connects it to classical integer sequences, expanding understanding of sequence transformations.

Findings

The operator changes zeros of characteristic polynomials in specific ways.

A subset of these operators forms a mathematical group.

Many relations involving Fibonacci, Catalan, and triangular numbers are derived.

Abstract

In this paper we study the action of a generalization of the Binomial interpolated operator on the set of linear recurrent sequences. We find how the zeros of characteristic polynomials are changed and we prove that a subset of these operators form a group, with respect to a well-defined composition law. Furthermore, we study a vast class of linear recurrent sequences fixed by these operators and many other interesting properties. Finally, we apply all the results to integer sequences, finding many relations and formulas involving Catalan numbers, Fibonacci numbers, Lucas numbers and triangular numbers.