Related papers: On the binomial convolution of arithmetical functi…

This note provides truncated formulae with explicit error terms to compute Euler products over primes in arithmetic progressions of rational fractions. It further provides such a formula for the product of terms of the shape $F(1/p, 1/p^s)$…

We consider binomial and inverse binomial sums at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi$ or $\log(2)$. In order to perform these simplifications, we view the series as specializations of…

Let $G(g;x):=\sum_{n\leq x}g(n)$ be the summatory function of an arithmetical function $g(n)$. In this paper, we prove that we can write weighted averages of an arbitrary fixed number $N$ of arithmetical functions $g_{j}(n),\,j\in\left\{…

A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…

Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted $\alext$ and is a Banach space under…

The group $Diff$ of diffeomorphisms of the circle is an infinite dimensional analog of the real semisimple Lie groups $U(p,q)$, $Sp(2n,R)$, $SO^*(2n)$; the space $\Xi$ of univalent functions is an analog of the corresponding classical…

In this paper we establish a new formula for the arithmetic functions that verify $ f(n) = \sum_{d|n} g(d)$ where $g$ is also an arithmetic function. We prove the following identity, $$\forall n \in \mathbb{N}^*, \ \ \ f(n) = \sum_{k=1}^n…

In this article we study some properties of the discrete convolution of Liouville function $S(n):=\sum_{m_{1}+m_{2}=n}\lambda\left(m_{1}\right)\lambda\left(m_{2}\right)$, which is a Goldbach-type counting function of representations. In…

Hoffstein and Hulse recently introduced the notion of shifted convolution Dirichlet series for pairs of modular forms $f_1$ and $f_2$. The second two authors investigated certain special values of symmetrized sums of such functions, numbers…

We present an approach to classical definitions and results on cumulant--moment relations and Wick polynomials based on extensive use of convolution products of linear functionals on a coalgebra. This allows, in particular, to understand…

The concept of a $1$-rotational factorization of a complete graph under a finite group $G$ was studied in detail by Buratti and Rinaldi. They found that if $G$ admits a $1$-rotational $2$-factorization, then the involutions of $G$ are…

Conical functions appear in a large number of applications in physics and engineering. In this paper we describe an extension of our module CONICAL for the computation of conical functions. Specifically, the module includes now a routine…

We consider a wide class of summatory functions F{f;N,p^m}=\sum_{k\leq N}f(p^m k), m\in \mathbb Z_+\cup {0}, associated with the multiplicative arithmetic functions f of a scaled variable k\in \mathbb Z_+, where p is a prime number.…

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)^+), the double tensor algebra of B, with the…

We investigate the relations between the rings ${\bf E}$, ${\bf G}$ and ${\bf D}$ of values taken at algebraic points by arithmetic Gevrey series of order either $-1$ ($E$-functions), $0$ (analytic continuations of $G$-functions) or $1$…

This is the central article of a series of three papers on cross product bialgebras. We present a universal theory of bialgebra factorizations (or cross product bialgebras) with cocycles and dual cocycles. We also provide an equivalent…

Convolution is a ubiquitous operation in mathematics and computing. The Kripke semantics for substructural and interval logics motivates its study for quantale-valued functions relative to ternary relations. The resulting notion of…

We establish a novel connection between the central binomial coefficients $\binom{2n}{n}$ and Gould's sequence through the construction of a specialized multivariate polynomial quotient ring. Our ring structure is characterized by ideals…

We present an explicit product formula for the spherical functions of the compact Gelfand pairs $(G,K_1)= (SU(p+q), SU(p)\times SU(q))$ with $p\ge 2q$, which can be considered as the elementary spherical functions of one-dimensional…

We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…