Factorization theorems and canonical representations for generating functions of special sums

TL;DR

This paper investigates matrix-based factorizations of generating functions for special sum sequences, proposing a canonical form that maximizes qualitative and quantitative properties, and explores conjectures on optimal factorizations.

Contribution

It introduces a new framework for defining canonical matrix factorizations of generating functions based on correlation statistics, extending known results for Lambert series.

Findings

Proposes a quantitative criterion for canonical factorizations

Establishes a connection between algebraic structures and optimal factorizations

Poses conjectures on maximal and minimal correlation-based factorizations

Abstract

This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. The last primary (non-appendix) section of the thesis explores the topic of how to best rigorously define a so-termed ``\emph{canonically best}'' matrix based factorization for a given class of convolution sum sequences. The notion of a canonical factorization for the generating function of such sequences needs to match the qualitative properties we find in the factorization theorems for Lambert series generating functions (LGFs). The expected qualitatively most expressive expansion we find in the LGF case results naturally from algebraic constructions of the underlying LGF series type. We propose a precise quantitative requirement to generalize this notion in terms of optimal cross-correlation statistics for certain sequences that define the matrix based factorizations of the generating function expansions we study. We finally pose a few conjectures on the types of matrix factorizations we expect to find when we are able to attain the maximal (respectively minimal) correlation statistic for a given sum type.