A theory of theta functions to the quintic base

TL;DR

This paper develops a comprehensive theory of quintic theta functions, establishing their properties, identities, and modular relations, and applies these to derive congruences for combinatorial functions like the partition function.

Contribution

It introduces a new framework for quintic theta functions, paralleling classical theory, and uses it to analyze modular forms and derive congruences for combinatorial functions.

Findings

Quintic theta functions satisfy analogues of classical identities.

A formal technique for congruences modulo powers of five is developed.

Connections between quintic theta functions, Eisenstein series, and differential equations are established.

Abstract

Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and counterparts of Jacobi's Principles of Duplication, Dimidiation and Change of Sign Formulas. The resulting library of quintic transformation formulas is used to describe series multisections for modular forms in terms of simple matrix operations. These efforts culminate in a formal technique for deducing congruences modulo powers of five for a variety of combinatorial generating functions, including the partition function. Further analysis of the quintic theta functions is undertaken by exploring their modular properties and their connection to Eisenstein series. The resulting relations lead to a coupled system of differential equations for the quintic theta functions.