Hypertranscendence of solutions of linear difference equations

Abstract: The general question of this talk is the classification of differentially algebraic solutions of linear difference equations of the following type:

$$ (\ast) \qquad a_0(z)f(z) + a_1(z)f(R(z)) + ... + a_n(z)f(R_n(z))=0$$

where, for every $i$, $a_i(z) \in \C(z)$, $R(z) \in \C(z)$ and $R_n(z)$ is the $n$-th composition of $R(z)$ with itself. We say that such a function is differentially algebraic over $\C(z)$ if there exist an non-zero integer $n$ and a non-zero polynomial $P \in \C(z)[X_0,..., X_n]$ such that $P(f(z),..., f^{(n)}(z))=0$, where $f^{(i)}$ is the $i$-th derivative of $f$ with respect to $z$. Otherwise, it is hypertranscendental over $\C(z)$.

The classification of differentially algebraic solutions is known for three types of non-linear difference equations : the Schröder's, Böttcher's and Abel's equations : $f(qz)=R(f(z))$, $f(z^d)=R(f(z))$, $f(R(z))=f(z)+1$, respectively, where $q \in \C^{\ast}$, $d \in \N$, $d \geq 2$. A classification of the differential algebraicity of solutions of linear difference equations of the above type $(\ast)$ is made in an article of B. Adamczewski, T. Dreyfus, C. Hardouin, for these same operators : q-differences $z \to qz$, mahlerian $z \to z^d$, and shift $z \to z+1$, by the means of an adapted difference Galois theory.

In this talk, we discuss the generalisation of these results to any function $R$ (rational or algebraic over $\C(z)$), in the case where $(\ast)$ is of order $1$. This is a work in progress with L. Di Vizio. Natural applications appear in examples of generating series of random walks, which satisfy this kind of equation of order $1$.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar