D-algebraic Guessing
Bertrand Teguia Tabuguia
TL;DR
This work develops a systematic framework for guessing $D$-algebraic ($D$-algebraic) equations from finite data for both generating functions and sequences, extending beyond traditional $D$-finite methods. It introduces structured algorithms that search for polynomial differential and difference equations of bounded degree and order, leveraging modular arithmetic over finite fields to manage large coefficients and linking differential monomials to power-series recurrences. The authors validate the approach through diverse examples, including the generating function for $\zeta(2n+2)$, subsequences of Catalan numbers, a two-dimensional differential tree automaton, and data from OEIS, while providing a Maple NLDE package to implement the methods. They also show a non-$D$-algebraicity result for the odd-prime subsequences using deep number-theoretic results (Maynard–Tao), illustrating both the reach and limits of the guessing framework. Overall, the work broadens the computational toolkit for nonlinear algebraic relations in combinatorics and number theory, offering a practical guess-and-prove paradigm that complements existing $D$-finite theory.
Abstract
Given finitely many consecutive terms of an infinite sequence, we discuss the construction of a polynomial difference equation that the sequence may satisfy. We also present a method to seek a candidate polynomial differential equation for its generating function. It appears that these methods often lead to effective D-algebraic operations.