Monoidal Alphabets for Generalized Harmonic Sums

TL;DR

This paper introduces a unified algebraic framework using monoidal alphabets to systematically analyze and generate identities for various classes of Euler-type sums and harmonic numbers.

Contribution

It develops a general finite-alphabet monoidal framework that encompasses classical and generalized harmonic sums, enabling systematic reduction and identity derivation.

Findings

Framework recovers many known Euler-sum identities.

Framework produces new identities in a uniform manner.

Nested sums reduce to finite harmonic-number objects under the framework.

Abstract

We develop a general finite-alphabet framework for Euler-type sums based on the notion of a monoidal alphabet. An alphabet of summand letters is called monoidal when it is closed under pointwise multiplication, thereby inducing the usual stuffle, or quasi-shuffle, algebra on the associated nested sums. This viewpoint places classical multiple harmonic numbers, colored harmonic sums, and several generalized Euler sums under a common structural mechanism. We focus on three fundamental families of monoidal alphabets: the ordinary power alphabet generated by $n$, the affine alphabet generated by linear factors $an+b$, and the polynomial-base alphabet generated by polynomial factors $P(n)$. The resulting classes of multiple harmonic numbers, multiple affine harmonic numbers, and multiple polynomial-base harmonic numbers provide systematic containers for a wide range of finite and infinite Euler-type sums. We prove closure and lifting results showing that nested sums whose summands are built from these alphabets, possibly multiplied by harmonic-number factors, reduce to the corresponding finite harmonic-number objects. As consequences, the framework recovers many known Euler-sum identities and produces many new identities in a uniform way. While reduction to simpler functions remains a separate and often difficult problem, the monoidal-alphabet perspective provides a unified algebraic language for organizing, transforming, and extending harmonic-sum identities.