Selected non-holonomic functions in lattice statistical mechanics and enumerative combinatorics

TL;DR

This paper investigates the algebraic and non-holonomic nature of certain functions in lattice statistical mechanics and combinatorics, showing they reduce to algebraic functions modulo primes and exploring their potential representation as diagonals of rational functions.

Contribution

It demonstrates that a specific non-holonomic series related to the q=4 case in combinatorics reduces to algebraic functions modulo various primes, suggesting a new perspective on their structure.

Findings

Series reduces to algebraic functions modulo primes 2 to 19

The q=4 series is likely non-holonomic over the rationals

Raises the possibility of representing these functions as diagonals of rational functions

Abstract

We recall that the full susceptibility series of the Ising model, modulo powers of the prime 2, reduce to algebraic functions. We also recall the non-linear polynomial differential equation obtained by Tutte for the generating function of the q-coloured rooted triangulations by vertices, which is known to have algebraic solutions for all the numbers of the form $2 +2 \cos(j\pi/n)$, the holonomic status of the q= 4 being unclear. We focus on the analysis of the q= 4 case, showing that the corresponding series is quite certainly non-holonomic. Along the line of a previous work on the susceptibility of the Ising model, we consider this q=4 series modulo the first eight primes 2, 3, ... 19, and show that this (probably non-holonomic) function reduces, modulo these primes, to algebraic functions. We conjecture that this probably non-holonomic function reduces to algebraic functions modulo (almost) every prime, or power of prime numbers. This raises the question to see whether such remarkable non-holonomic functions can be seen as ratio of diagonals of rational functions, or algebraic, functions of diagonals of rational functions.