Related papers: Criteria for apwenian sequences

The Hankel transform H_n[f(x)](q) = int_0^infinity xf(x)J_n(qx)dx is studied for integer n>=-1 and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of…

It is well known that the derangement numbers $d_n$, which count permutations of length $n$ with no fixed points, satisfy the recurrence $d_n=nd_{n-1}+(-1)^n$ for $n\ge1$. Combinatorial proofs of this formula have been given by Remmel,…

For $N \geq 2$, we study the structure of definable abelian group extensions of the additive group $(\mathbb{R}^N,+)$ by countable abelian (Borel) groups $G$. Given an extension $H$ of $(\mathbb{R}^N,+)$ by $G$, we measure the definability…

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…

Let $p$ be a prime number and consider a $p$-automatic sequence ${\bf u}=(u_{n})_{n\in\N}$ and its generating function $U(X)=\sum_{n=0}^{\infty}u_{n}X^{n}\in\mathbb{F}_{p}[[X]]$. Moreover, let us suppose that $u_{0}=0$ and $u_{1}\neq 0$ and…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several interesting identities involving this sequence such as $F_n^2+F_{n+1}^2=F_{2n+1}$, for all $n\geq…

This article analyzes F\olner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70ies. We prove that each essentially hyponormal operator has a proper…

A $K$-Fibonacci sequence is a binary recurrence sequence where $F_0=0$, $F_1=1$, and $F_n=K\cdot F_{n-1}+F_{n-2}$. These sequences are known to be periodic modulo every positive integer greater than $1$. If the length of one shortest period…

We prove supercongruences modulo $p^2$ for values of truncated hypergeometric series at some special points. The parameters of the hypergeometric series are $d$ copies of $1/2$ and $d$ copies of $1$ for any integer $d\ge2$.

We establish two structural results for Moore homology of ample groupoids. First, for every ample groupoid $\mathcal{G}$ and every discrete abelian coefficient group $A$, we prove a universal coefficient theorem relating the homology groups…

We prove that for every positive integer $m$, there exist infinitely many simple abelian varieties over $\mathbb{F}_2$ of order $m$. The method is constructive, building on the work of Madan--Pal in the case $m=1$ to produce an explicit…

We prove that the uniform recurrence of morphic sequences is decidable. For this we show that the number of derived sequences of uniformly recurrent morphic sequences is bounded. As a corollary we obtain that uniformly recurrent morphic…

We give a criterion for bounding the homological finiteness length of certain HF-groups. This is used in two distinct contexts. Firstly, the homological finiteness length of a non-uniform lattice on a locally finite n-dimensional…

Let $E$ be a nonsingular elliptic curve over the rational numbers, and let $\tau^n=p^n+1-\#E(\mathbb{F}_{p^n})$. A result in the current literature claims that the normalized traces of Frobenius…

Unstable separatrix solutions for the first and second Painlev\'e transcendents are studied both numerically and analytically. For a fixed initial condition, say $y(0)=0$, there is a discrete set of initial slopes $y'(0)=b_n$ that give rise…

In a companion paper \cite{jon-fei}, we established asymptotic formulae for the joint moments of derivatives of the characteristic polynomials of CUE random matrices. The leading order coefficients of these asymptotic formulae are expressed…

Generalized abelian equivalence compares words by their factors up to a certain bounded length. The associated complexity function counts the equivalence classes for factors of a given size of an infinite sequence. How practical is this…

The asynchronous automaton associated with a Boolean network $f:\{0,1\}^n\to\{0,1\}^n$ is considered in many applications. It is the finite deterministic automaton with set of states $\{0,1\}^n$, alphabet $\{1,\dots,n\}$, where the action…

For every monic polynomial $f \in \mathbb{Z}[X]$ with $\operatorname{deg}(f) \geq 1$, let $\mathcal{L}(f)$ be the set of all linear recurrences with values in $\mathbb{Z}$ and characteristic polynomial $f$, and let \begin{equation*}…

We study the recursions $A(n) = A(n-a-A^k(n-b)) + A(A^k(n-b))$ where $a \geq 0$, $b \geq 1$ are integers and the superscript $k$ denotes a $k$-fold composition, and also the recursion $C(n) = C(n-s-C(n-1)) + C(n-s-2-C(n-3))$ where $s \geq…