Related papers: Becker's conjecture on Mahler functions

We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion…

A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…

In $1735$ Euler \cite{1} proved that for each positive integer $k$, the series $\zeta(2k) = \sum_{\ell=1}^{\infty} \ell^{-2k}$ converges to a rational multiple of $\pi^{2k}$. Many demonstrations of this fact are now known, and Euler's…

Let $\mathcal{A}$ be the family of functions $f(z)=z+a_2z^2+...$ which are analytic in the open unit disc $\mathbb{D}=\{z: |z|<1 \}$, and denote by $\pe$ of functions $p(z)=z+p_1z+p_2z^2+...$ analytic in $\de$ such that $p(z)$ is in $\pe$…

Let K be a number field, let f(x) in K(x) be a rational function of degree d> 1, and let z in K be a wandering point such that f^n(z) is nonzero for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many…

Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant $\gamma$ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's…

In this paper we show that if an entire function $f(z_1,z_2)$ of two (or more) complex variables verifies $\norm{f(z_1,z_2)}\leq K(\norm{P(z_1,z_2)})$, where $P(z_1,z_2)$ is a polynomial that is not a power in $\CC[[z_1,z_2]]$, and $K$ is…

This paper mainly uses the nonnegative continuous function $\{\zeta_n(r)\}_{n=0}^{\infty}$ to redefine the Bohr radius for the class of analytic functions satisfying $\real f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of…

We prove that every Mahler series, over a field of characteristic $0$, with multiplicative coefficients is regular in the sense of Allouche and Shallit. We also obtain an explicit characterization of such series. This yields a joint…

In 2007, A.I.Aptekarev and his collaborators discovered a sequence of rational approximations to Euler's constant $\gamma$ defined by a linear recurrence. In this paper, we generalize this result and present an explicit construction of…

We show that a polyregular word-to-word function is regular if and only if its output size is at most linear in its input size. Moreover a polyregular function can be realized by: a transducer with two pebbles if and only if its output has…

A word-to-word function is rational if it can be realized by a non-deterministic one-way transducer. Over finite words, it is a classical result that any rational function is regular, i.e. it can be computed by a deterministic two-way…

A real-valued function on R^n is k-regulous, where k is a nonnegative integer, if it is of class C^k and can be represented as a quotient of two polynomial functions on R^n. Several interesting results involving such functions have been…

We investigate semiconjugate rational functions, that is rational functions $A,$ $B$ related by the functional equation $A\circ X=X\circ B$, where $X$ is a rational function of degree at least two. We show that if $A$ and $B$ is a pair of…

Consider an open, bounded set $\Omega\subset \mathbb{C}$, a positive integer $k$ and a compact $\mathcal{K}\subset \Omega$ of cardinality strictly greater than $k$. We prove that, for any function $f$ which is holomorphic in $\overline…

We give new closed and explicit formulas for "multiple zeta values" at non-positive integers of generalized Euler-Zagier multiple zeta-functions. We first prove these formulas for a small convenient class of these multiple zeta-functions…

We present an algorithm producing all rational functions $f$ with prescribed $n+1$ Taylor coefficients at the origin and such that $\|f\|_\infty\le 1$ and $\deg f\le k$ for every fixed $k\ge n$. The case where $k<n$ is also discussed.

The free Baker-Akhiezer modules on rational varieties obtained from ${\mathbb C}P^{1}\times{\mathbb C}P^{n-1}$ by identification of two hypersurfaces are constructed. The corollary of this construction is the existence of embedding of…

This paper investigates the location of the zeros of a sequence of polynomials generated by a rational function with a binomial-type denominator. We show that every member of a two-parameter family consisting of such generating functions…

We establish precise relations between Euler systems that are respectively associated to a $p$-adic representation $T$ and to its Kummer dual $T^*(1)$. Upon appropriate specialization of this general result, we are able to deduce the…