Related papers: More Infinite Products: Thue-Morse and the Gamma f…
We study the behavior of a polynomial sequence which is defined by iterating a polynomial pair under Thue-Morse dynamic. We show that in suitable sense, the sequence will behave like $\{2\cos 2^nx: n\ge 1\}$. Basing on this property we can…
We derive the infinite product of the tangent function expressed in terms of trigonometric expressions such as Eulers Sinc function and Vietes formula, along with their generalizations. All the results presented in this work are novel.
Continued fraction expansions and Hankel determinants of automatic sequences are extensively studied during the last two decades. These studies found applications in number theory in evaluating irrationality exponents. The present paper is…
We consider products of $q$-gamma functions with rational arguments, and prove several $q$-generalizations of recent works concerning products of gamma functions. In particular, we consider products indexed by Dirichlet characters, and…
The binomial convolution of two sequences $\{a_n\}$ and $\{b_n\}$ is the sequence whose $n$th term is $\sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$. If $\{a_n\}$ and $\{b_n\}$ have rational generating functions then so does their binomial…
The Thue-Morse sequence (1, -1, -1, 1, -1, 1, 1, ...) can in a sense be naturally extended to a continuous function f called the Fabius function. It is shown how to determine the exact value of f(x) whenever x is the ratio between a…
In the $1970$s, Nicolas proved that the partition function $p(n)$ is log-concave for $ n > 25$. In \cite{HNT21}, a precise conjecture on the log-concavity for the plane partition function $\func{pp}(n)$ for $n >11$ was stated. This was…
Studying and comparing arithmetic properties of a given automatic sequence and the sequence of coefficients of the composition inverse of the associated formal power series (the formal inverse of that sequence) is an interesting problem.…
It is a classical fact that the exponential function is solution of the integral equation $ \int_0^X f(x)dx + f(0) =f(X)$. If we slightly modify this equation to $ \int_0^X f(x)dx+f(0)=f(\alpha X)$ with $\alpha\in ]0,1[$, it seems that no…
Let $I_n(x)=\prod_{i=1}^n \left( 1+x^{F_{i+1}}\right)$, where $F_{i+1}$ denotes a Fibonacci number. Let $v_r(n)$ denote the sum of the $r$th powers of the coefficients of $I_n(x)$. Our prototypical result is that $\sum_{n\geq 0} v_2(n)x^n=…
In this paper we study the privileged complexity function of the Thue-Morse word. We prove a recursive formula describing this function, and using the formula we show that the function is unbounded and that the values of the function have…
Product identities in two variables $x, q$ expand infinite products as infinite sums, which are linear combinations of theta functions; famous examples include Jacobi's triple product identity, Watson's quintuple identity, and Hirschhorn's…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
In this paper we study sums of Dirichlet series whose coefficients are terms of the Thue-Morse sequence and variations thereof. We find closed-form expressions for such sums in terms of known constants and functions including the Riemann…
In this note, we evaluate a multivariable family of infinite products which generalize Guillera's infinite product for $e$, and Ser's formula (rediscovered by Sondow) for $e^\gamma$. We describe formulas for the products in terms of special…
We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) <= n for n >= 4 and that f(n) = n infinitely…
I present and analyze a quadratically convergent algorithm for computing the infinite product \prod_{n=1}^\infty (1 - tx^n) for arbitrary complex t and x satisfying |x| < 1, based on the identity \prod_{n=1}^\infty (1 - tx^n) =…
We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of "inflating" the dimensions, we recover the classical expressions for realization of a regular product of rational…
If $p$ is a prime number, consider a $p$-automatic sequence $(u_n)_{n\ge 0}$, and let $U(X) = \sum_{n\ge 0} u_n X^n \in \mathbb{F}_p[[X]]$ be its generating function. Assume that there exists a formal power series $V(X) = \sum_{n\ge 0} v_n…
Given an integer $q\ge 2$ and a real number $c\in [0,1)$, consider the generalized Thue-Morse sequence $(t_n^{(q;c)})_{n\ge 0}$ defined by $t_n^{(q;c)} = e^{2\pi i c S_q(n)}$, where $S_q(n)$ is the sum of digits of the $q$-expansion of $n$.…