A semigroup approach to iterated binomial transforms

TL;DR

This paper introduces a semigroup framework for iterated binomial transforms on sequences, providing explicit formulas, inverse operations, and applications to classical recurrence families, revealing a root-shift property.

Contribution

It develops a unified semigroup approach to binomial transforms, including their inverse and root-shift behavior, and applies it to classical recurrence sequences.

Findings

The binomial transform forms an additive semigroup with an explicit inverse.

Applying the transform shifts the roots of recurrence characteristic equations.

Classical sequences like Fibonacci and Lucas are uniformly analyzed within this framework.

Abstract

We study a one-parameter family of binomial-convolution operators acting on sequences. These operators form an additive semigroup with an explicit inverse, and they subsume iterated classical binomial transforms as a special case. We describe the action in terms of ordinary and exponential generating functions, interpret the transform in the Riordan-array framework, and prove a general root-shift principle for constant-coefficient linear recurrences: applying the transform shifts the characteristic roots by a fixed amount. Several classical families (Fibonacci, Lucas, Pell, Jacobsthal, Mersenne) are treated uniformly as illustrative examples.