On the exponential generating function of labelled trees

The lambda extensions of the Ising correlation functions C(M,N)

“…Let us write (34) as 5 A D-finite function is a function solution of a linear ODE with polynomial coefficients. A DD-finite function is a function solution of a linear differential equation whose coefficients are D-finite functions [9]. 6 A differentially algebraic function [11] is a function f (t) solution of a polynomial relation…”

“…In [1] it has been underlined that these correlation functions C(N, N) have lambda extensions which are also solutions of (3), that can be defined using a 'form factor' low-temperature expansion [1,7] (see (9) in [1]):…”

Differentially definable functions: a survey

“…The package dd_functions [5] provides a full implementation of differentially definable functions in SageMath [13]. It allows the user to define the rings D(R) for a given differential ring R, to build functions using the data structure (2), and it provides a user friendly interface for computing closure properties.…”

“…The natural question arises, whether all differentially algebraic functions are D n -finite for some n ∈ N. The answer to this question is no. A simple counter example has been given by Noordman, van der Put, and Top [12]: the solutions to the non-linear differential equation y ′ = y 3 − y 2 are not D n -finite for any n. A combinatorial example that can be reduced to this case has been investigated in [2].…”

“…For D-finite functions, there are algorithms for certified arbitrary precision numerical evaluation [11] at any regular point only using the defining differential equation, i.e., the data structure (2). These algorithms require a pre-processing step: the computation of the singularities of the differential equation (1).…”