Diagonals of rational functions, pullbacked 2F1 hypergeometric functions and modular forms (unabrigded version)

TL;DR

This paper demonstrates that diagonals of certain rational functions can be expressed as pullbacked 2F1 hypergeometric functions, revealing their modular form nature and extending these results to more complex families with additional parameters.

Contribution

It introduces a family of rational functions with diagonals expressed as pullbacked 2F1 hypergeometric functions, establishing their modular form properties and generalizing to higher parameter families.

Findings

Diagonals of specific rational functions are expressible as pullbacked 2F1 hypergeometric functions.

Some diagonals are shown to be modular forms via algebraically related rational function pullbacks.

Extension of results to families with more parameters and added cubic terms.

Abstract

We recall that diagonals of rational functions naturally occur in lattice statistical mechanics and enumerative combinatorics. We find that a seven-parameter rational function of three variables with a numerator equal to one (reciprocal of a polynomial of degree two at most) can be expressed as a pullbacked 2F1 hypergeometric function. This result can be seen as the simplest non-trivial family of diagonals of rational functions. We focus on some subcases such that the diagonals of the corresponding rational functions can be written as a pullbacked 2F1 hypergeometric function with two possible rational functions pullbacks algebraically related by modular equations, thus showing explicitely that the diagonal is a modular form. We then generalise this result to eight, nine and ten parameters families adding some selected cubic terms at the denominator of the rational function defining the diagonal. We finally show that each of these previous rational functions yields an infinite number of rational functions whose diagonals are also pullbacked 2F1 hypergeometric functions and modular forms.