On the binomial convolution of arithmetical functions

对称函数的新组合恒等式

The new combinatorial identities of symmetric functions

本文利用指数生成函数等初等方法，研究了与对称函数相关的二项式卷积及Bernoulli、Euler和Genocchi数与多项式的卷积和，推导并建立了二元多项式及相关特殊多项式的多个新组合恒等式。

关于二项式定理与离散卷积之间的联系

On the link between binomial theorem and discrete convolution

本文引入多项式$\mathbf{P}^{m}_{b}(x)$并建立奇数次幂的多项式恒等式，借此揭示了二项式定理与奇数次幂离散卷积的联系，并将其推广至多项式情形，最后通过Mathematica程序验证。

Convolution identities for Tribonacci numbers via the diagonal of a bivariate generating function

Convolutions for Tribonacci numbers involving binomial coefficients are treated with ordinary generating functions and the diagonalization method of Hautus and Klarner. In this way, the relevant generating function can be established, which is rational. The coefficients can also be expressed. It is sketched how to extend this to Tetranacci numbers and similar quantities.

On the operations of sequences in rings and binomial type sequences

Given a commutative ring with identity $R$, many different and interesting operations can be defined over the set $H_R$ of sequences of elements in $R$. These operations can also give $H_R$ the structure of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well--known transforms of sequences (such as the Stirling trans…

On the error term in a Parseval type formula in the theory of Ramanujan expansions

Given two arithmetical functions $f,g$ we derive, under suitable conditions, asymptotic formulas with error term, for the convolution sums $\sum_{n \le N} f(n) g(n+h)$, building on an earlier work of Gadiyar, Murty and Padma. A key role in our method is played by the theory of Ramanujan expansions for arithmetical functions.

Inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs

Let $S=\{x_1,x_2,\dots,x_n\}$ be a set of distinct positive integers, and let $f$ be an arithmetical function. The GCD matrix $(S)_f$ on $S$ associated with $f$ is defined as the $n\times n$ matrix having $f$ evaluated at the greatest common divisor of $x_i$ and $x_j$ as its $ij$ entry. The LCM matrix $[S]_f$ is defined similarly. We consider inertia, positive definiteness and $\ell_p$ norm of GCD and LCM matrices and their unitary analogs. Proofs are based on matrix factorizations and convolutions of arithmetical functions.

A convolution for the complete and elementary symmetric functions

In this paper we give a convolution identity for the complete and elementary symmetric functions. This result can be used to proving and discovering some combinatorial identities involving $r$-Stirling numbers, $r$-Whitney numbers and $q$-binomial coefficients. As a corollary we derive a generalization of the quantum Vandermonde's convolution identity.

On a class of arithmetic convolutions involving arbitrary sets of integers

Let $d,n$ be positive integers and $S$ be an arbitrary set of positive integers. We say that $d$ is an $S$-divisor of $n$ if $d|n$ and gcd $(d,n/d)\in S$. Consider the $S$-convolution of arithmetical functions given by (1.1), where the sum is extended over the $S$-divisors of $n$. We determine the sets $S$ such that the $S$-convolution is associative and preserves the multiplicativity of functions, respectively, and discuss other basic properties of it. We give asymptotic formulae with error terms for the functions $σ_S(n)$ and $τ_S(n)$, representing the sum and the number of $S$-divisors of $…

An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function

Given a periodic function $f$, we study the convergence almost everywhere and in norm of the series $\sum_{k} c_k f(kx)$. Let $f(x)= \sum_{m=1}^\infty a_m \sin {2πm x}$ where $\sum_{m=1}^\infty a_{m }^2d(m) <\infty$ and $d(m)=\sum_{d|m} 1$, and let $f_n(x) = f(nx)$. We show by using a new decomposition of squared sums that for any $K\subset \N$ finite, $ \|\sum_{k\in K} c_k f_k \|_2^2 \le ( \sum_{m=1}^\infty a_{m }^2 d(m) ) \sum_{k\in K } c_{k}^2d(k^2)$. If $f^s (x)= \sum_{j=1}^\infty \frac{\sin 2πjx}{j^s}$, $s>1/2$, by only using elementary Dirichlet convolution calculus, we show that for $0<…

On the error term in a Parseval type formula in the theory of Ramanujan expansions II

For two arithmetical functions $f$ and $g$ with absolutely convergent Ramanujan expansions, Murty and Saha have recently derived asymptotic formulas with error term for the convolution sum $\sum_{n \le N} f(n) g(n+h)$ under some suitable conditions (see http://arxiv.org/abs/1506.01945). In this follow up article we improve these results with a weakened hypothesis which is in some sense minimal.

On the Brousseau sums $\sum_{i=1}^n i^p F_i$

We start with new convolution formulas for $F_n - n^p$ involving only the binomial coefficients. Then, we use those to find direct formulas for the sums $\sum_{i=1}^n i^p F_{n-i}$ and $\sum_{i=1}^n i^p F_i$, and we show how our formulas connect to work in earlier papers by Ledin, Brousseau, Zeitlin, Adegoke, Shannon and Ollerton, and Kinlaw, Morris, and Thiagarajan.