Matrix Representation for Multiplicative Nested Sums

TL;DR

This paper introduces a matrix-based method to compute multiplicative nested sums, generalizing harmonic sums and relating them to stochastic processes and combinatorial identities.

Contribution

It presents a novel matrix representation approach for multiplicative nested sums, enabling easier derivation of relations and identities.

Findings

Matrix multiplication efficiently computes nested sums.

Special cases relate to stochastic transition matrices.

Derivation of combinatorial identities from matrix identities.

Abstract

We study multiplicative nested sums, which are generalizations of harmonic sums, and provide a calculation through multiplication of index matrices. Special cases interpret the index matrices as stochastic transition matrices of random walks on a finite number of sites. Relations among multiplicative nested sums, which are generalizations of relations between harmonic series and multiple zeta functions, can be easily derived from identities of the index matrices. Combinatorial identities and their generalizations can also be derived from this computation.