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

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.

27 pages

References in corpus (7)

• Computing periods of rational integrals
• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Automatic congruences for diagonals of rational functions
• A Fast Algorithm for Computing the p-Curvature
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity
• Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version)

Cited by in corpus (4)

• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• Is the full susceptibility of the square-lattice Ising model a differentially algebraic function?
• Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)
• The lambda extensions of the Ising correlation functions C(M, N)