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Équations différentielles motiviques et au–delà

On deciding transcendence of D-finite power series


A formal power series in Q[[t]] is said to be D-finite ("differentially finite"), or holonomic, if it satisfies a linear differential equation with polynomial coefficients. D-finite power series are ubiquitous in number theory and combinatorics. 

In a seminal article (1980), Richard Stanley asked whether it possible to decid if a given D-finite power series is algebraic or transcendental.
Several very useful sufficient criteria for transcendence exist, e.g.,using asymptotics, but none of them is also a sufficient condition. 

Characterizing the transcendence of a D-finite power series is highly nontrivial even if its coefficient sequence satisfies a recurrence of. first order: this question was completely solved only in 1989 by Frits Beukers and Gert Heckman. In this talk, I will present answers to
Stanley’s question and illustrate them through several examples coming from number theory and combinatorics.