Related papers: On an arithmetic convolution
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, $$\forall n \in \mathbb{N}^*, \ \ \ f(n) = \sum_{k=1}^n…
We give a new characterization of the set $\mathcal{C}$ of Carmichael numbers in the context of $p$-adic theory, independently of the classical results of Korselt and Carmichael. The characterization originates from a surprising link to the…
A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for…
We use the correlation functions of vertex operators to give a proof of Cauchy's formula \begin{align*} \prod^K_{i=1}\prod^N_{j=1}(1-x_iy_j)=\sum_{\mu\subseteq [K\times N]}(-1)^{|\mu|}s_{\mu}\{x\}s_{\mu'}\{y\}. \end{align*} As an…
It is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a…
Let $H$ and $K$ be locally compact groups and $\tau:H\to Aut(K)$ be a continuous homomorphism and also let $G_\tau=H\ltimes_\tau K$ be the semi-direct product of $H$ and $K$ with respect to $\tau$. We define left and also right…
Using general identities for difference operators, as well as a technique of symbolic computation and tools from probability theory, we derive very general kth order (k \ge 2) convolution identities for Bernoulli and Euler polynomials. This…
We present a unified algebraic framework utilizing the formal Bell transform to bridge the Dirichlet convolution of arithmetic functions with the combinatorial structure of infinite Euler-type products. By analyzing the logarithmic…
We study the convolution function $$ C[f(x)] := \int_1^x f(y)f({x\over y}) {{\rm d} y\over y} $$ when $f(x)$ is a suitable number-theoretic error term. Asymptotics and upper bounds for $C[f(x)]$ are derived from mean square bounds for…
We continue our study of convolution sums of two arithmetical functions $f$ and $g$, of the form $\sum_{n \le N} f(n) g(n+h)$, in the context of heuristic asymptotic formul\ae. Here, the integer $h\ge 0$ is called, as usual, the {\it shift}…
Given modular forms $f$ and $g$ of weights $k$ and $\ell$, respectively, their Rankin-Cohen bracket $[f,g]^{(k, \ell)}_n$ corresponding to a nonnegative integer $n$ is a modular form of weight $k +\ell +2n$, and it is given as a linear…
In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman-Opdam hypergeometric functions of type $BC$. For specific discrete series of multiplicities these hypergeometric…
The functional equation f(p(z))=g(q(z)) is studied, where p,q are polynomials and f,g are trancendental meromorphic functions in C. We find all the pairs p,q for which there exist nonconstant f,g satisfying our equation and there exist no…
Let K be a complete algebraically closed p-adic field of characteristic zero. Let f, g be two transcendental meromorphic functions in the whole field K or meromorphic functions in an open disk that are not quotients of bounded analytic…
In this paper, we study sums of shifted products $\sum\limits_{n \leq x} F(n) G(n-h)$ for any $|h| \leq x/2$ and arithmetic functions $F=f*1$ and $G=g*1$, with $f$ and $g$ small. We obtain asymptotic formula for different orders of…
We discuss a product formula for $F$-polynomials in cluster algebras, and provide two proofs. One proof is inductive and uses only the mutation rule for $F$-polynomials. The other is based on the Fock-Goncharov decomposition of mutations.…
In this paper we offer a definition of monogenicity for functions defined on $\rr^{n+1}$ with values in the Clifford algebra $\rr_n$ following an idea inspired by the recent papers \cite{gs}, \cite{advances}. This new class of monogenic…
We give estimates for the convolution product of an arbitrary number of endlessly continuable functions. This allows us to deal with nonlinear operations for the corresponding resurgent series, e.g. substitution into a convergent power…
A function of positive type can be defined as a positive functional on a convolution algebra of a locally compact group. In the case where the group is abelian, by Bochner's theorem a function of positive type is, up to normalization, the…
The $t=0$ specialization of the Mimachi-Noumi Cauchy-type identity rewrites certain infinite product in terms of specialized nonsymmetric Macdonald polynomials of type $GL_n$. We interpret the infinite product as a character of the space of…