Related papers: Transcendental Infinite Products Associated with t…
In this article we introduce a new approach to compute infinite products defined by automatic sequences involving the Thue-Morse sequence. As examples, for any positive integers $q$ and $r$ such that $0 \leq r \leq q-1$, we find infinitely…
Letting $(t_n)$ denote the Thue-Morse sequence with values $0, 1$, we note that the Woods-Robbins product $$ \prod_{n \geq 0} \left(\frac{2n+1}{2n+2}\right)^{(-1)^{t_n}} = 2^{-1/2} $$ involves a rational function in $n$ and the $\pm 1$…
Given an integer $q\ge2$ and $\theta_1,\cdots,\theta_{q-1}\in\{0,1\}$, let $(\theta_n)_{n\ge0}$ be the generalized Thue-Morse sequence, defined to be the unique fixed point of the morphism $$0\mapsto0\theta_1\cdots\theta_{q-1}$$…
We provide a non-trivial measure of irrationality for a class of Mahler numbers defined with infinite products which cover the Thue-Morse constant.
Let $(u_n)_{n\ge 0}$ denote the Thue-Morse sequence with values $\pm 1$. The Woods-Robbins identity below and several of its generalisations are well-known in the literature…
We study a class of infinite words $x_k$ , where $k$ is a positive integer, recently introduced by J. Shallit. This class includes the Thue-Morse sequence $x_1$, the Fibonacci-Thue-Morse sequence $x_2$, and the Allouche-Johnson sequence…
Using an application of Schmidt's Subspace Theorem, this paper gives new transcendence criteria for rapidly converging infinite products of algebraic numbers. The paper also improves existing criteria for irrationality of products and…
The celebrated Thue-Morse sequence, or the Prouhet-Thue-Morse sequence (A010060 in the OEIS), has a number of interesting properties and is a rich source to many (counter)examples. We introduce two different square-free sequences on three…
Touchard-Riordan-like formulas are some expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove these kind of formulas, related with integer…
Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This goes back to Euler. A purpose of this note…
The classic Thue--Morse measure is a paradigmatic example of a purely singular continuous probability measure on the unit interval. Since it has a representation as an infinite Riesz product, many aspects of this measure have been studied…
An infinite $\pm 1$-sequence is called {\it Apwenian} if its Hankel determinant of order $n$ divided by $2^{n-1}$ is an odd number for every positive integer $n$. In 1998, Allouche, Peyri\`ere, Wen and Wen discovered and proved that the…
Taking the product of (2n+1)/(2n+2) raised to the power +1 or -1 according to the n-th term of the Thue-Morse sequence gives rise to an infinite product P while replacing (2n+1)/(2n+2) with (2n)/(2n+1) yields an infinite product Q, where P…
Given a countable set X (usually taken to be the natural numbers or the integers), an infinite permutation \pi of X is a linear ordering of X. This paper investigates the combinatorial complexity of the infinite permutation on the natural…
Several infinite products are studied that satisfy the transformation relation of the type $f(\alpha)=f(1/\alpha)$. For certain values of the parameters these infinite products reduce to modular forms. Finite counterparts of these infinite…
In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The matrices in the product are taken from…
We prove the identity $$ \prod_{\gamma}\left(\frac{e^{l(\gamma)}+1}{e^{l(\gamma)}-1}\right)^{2h}=\exp\left(\frac{l_1+l_2+l_3}{2}\right), $$ (or $$…
The Thue--Morse sequence $\{t(n)\}_{n\geqslant 1}$ is the indicator function of the parity of the number of ones in the binary expansion of positive integers $n$, where $t(n)=1$ (resp. $=0$) if the binary expansion of $n$ has an odd (resp.…
In this paper, we investigate several infinite products with vanishing Taylor coefficients in arihmetic progressions. These infinite products are closely related to the Rogers--Ramanujan continued fraction. Moreover, a handful of new…
Let $p$ be a prime number and $\xi$ an irrational $p$-adic number. Its multiplicative irrationality exponent ${{\mu^{\times}}} (\xi)$ is the supremum of the real numbers ${{\mu^{\times}}}$ for which the inequality $$ |b \xi - a|_{p} \leq |…