Modular binomials with an application to periodic sequences

TL;DR

This paper introduces new recurrence relations for binomial coefficients modulo prime powers to analyze the primitives of modular periodic sequences, reducing complex cases to constant sequences and applying findings to a musical sequence.

Contribution

It develops novel recurrence relations for modular binomial coefficients and demonstrates their use in understanding the evolution of sequence primitives, including a musical application.

Findings

Recurrence relations for binomial coefficients modulo prime powers

Reduction of primitive sequence analysis to constant sequences

Application to a sequence from Vieru's 'Book of Modes'

Abstract

We study, through new recurrence relations for certain binomial coefficients modulo a power of a prime, the evolution of the primitives of a modular periodic sequence. We prove that we can reduce to study primitives of constant sequences and that the latter are controlled by modular binomial coefficients. Finally we apply our results to describe the dynamics of the primitives of the sequence considered by the Romanian composer Vieru in his "Book of Modes".