Multiple Sums Research Articles

Moment conditions in strong laws of large numbers for multiple sums and random measures

The validity of the strong law of large numbers for multiple sums Sn of independent identically distributed random variables Zk, k≤n, with r-dimensional indices is equivalent to the integrability of |Z|(log+|Z|)r−1, where Z is the generic summand. We consider the strong law of large numbers for more general normalizations, without assuming that the summands Zk are identically distributed, and prove a multiple sum generalization of the Brunk–Prohorov strong law of large numbers. In the case of identical finite moments of order 2q with integer q≥1, we show that the strong law of large numbers holds with the normalization (n1⋯nr)1∕2(logn1⋯lognr)1∕(2q)+ε for any ε>0.The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.

Using the “Freshman's Dream” to Prove Combinatorial Congruences

Recently, William Y.C. Chen, Qing-Hu Hou, and Doron Zeilberger developed an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences, namely those (like the Catalan and Motzkin sequences) that are expressible in terms of constant terms of powers of Laurent polynomials. We first give a leisurely exposition of their approach and then extend it in two directions. The Laurent polynomials may be of several variables, and instead of single sums we have multiple sums. In fact, we even combine these two generalizations. We conclude with some super-challenges.

Limits of Multiplicities in Excellent Filtrations and Tensor Product Decompositions for Affine Kac-Moody Algebras

We express the multiplicities of the irreducible summands of certain tensor products of irreducible integrable modules for an affine Kac-Moody algebra over a simply laced Lie algebra as sums of multiplicities in appropriate excellent filtrations (Demazure flags). As an application, we obtain expressions for the outer multiplicities of tensor products of two fundamental modules for $\widehat{\mathfrak{sl}}_2$ in terms of partitions with bounded parts, which subsequently lead to certain partition identities.

Improving the Existence Bounds for Grid-Block Difference Families

In this paper, by employing Weil's Theorem on multiplicative character sums, an intermediate algebraic consequence on the existence bound of an element satisfying certain cyclotomic conditions in a finite field is proposed. In many cases, this approach improves the bound due to Buratti and Pasotti (Finite Fields Appl 15(3), 332---344, 2009), which can be widely used for showing the asymptotic existence of combinatorial designs with point-regular automorphisms. Moreover, this approach is applied to improving the existence bound for grid-block difference families, which can be regarded as generalizations of difference families.

On almost everywhere convergence of lacunary sequences of multiple rectangular Fourier sums

Let a sequence of d-dimensional vectors n k = (n 1 , n 2 ,..., n ) with positive integer coordinates satisfy the condition n = α j m k +O(1), k ∈ ℕ, 1 ≤ j ≤ d, where α 1 > 0,..., α d > 0 and {m k } =1 ∞ is an increasing sequence of positive integers. Under some conditions on a function φ: [0,+∞) → [0,+∞), it is proved that, if the sequence of Fourier sums $${S_{{m_k}}}$$ (g, x) converges almost everywhere for any function g ∈ φ(L)([0, 2π)), then, for any d ∈ ℕ and f ∈ φ(L)(ln+ L) d−1([0, 2π) d ), the sequence $${S_{{n_k}}}$$ (f, x) of rectangular partial sums of the multiple trigonometric Fourier series of the function f and the corresponding sequences of partial sums of all conjugate series converge almost everywhere.

The group of multi-dimensional Riordan arrays

We first consider two groups, F0={g∈C[[z]]|g(0)≠0} under multiplication and F1=zF0 under composition, where C[[z]] is the ring of formal power series over the complex field. It is known that the Riordan group R is isomorphic to the semidirect product F0⋊F1. It may be viewed as a group extension of F1 by F0. In this paper, the group of three-dimensional Riordan arrays is obtained from an extension of the group R by F0. This concept extends to the group of multi-dimensional Riordan arrays. As an application, we illustrate the use of the three-dimensional Riordan array in multiple combinatorial sums.

Some q-congruences for homogeneous and quasi-homogeneous multiple q-harmonic sums

We show some new Wolstenholme type q-congruences for some classes of multiple q-harmonic sums of arbitrary depth with strings of indices composed of ones, twos, and threes. Most of these results are q-extensions of the corresponding congruences for ordinary multiple harmonic sums obtained by the authors in a previous paper. We also establish duality congruences for multiple q-harmonic non-strict sums and a kind of duality for multiple q-harmonic strict sums. Finally, we pose a conjecture concerning two kinds of cyclic sums of multiple q-harmonic sums.

On the integrality of the first and second elementary symmetric functions of $1, 1/2^{s_2}, ...,1/n^{s_n}$

It is well known that the harmonic sum $H_{n}(1)=\sum_{1\leq k\leq n}\frac{1}{k}$is never an integer for $n>1$. Erd\{o}s and Niven proved in 1946 thatthe multiple harmonic sum$H_{n}(\{1\}^r)=\sum_{1\leq k_{1}<\cdots< k_{r}\leq n}\frac{1}{k_{1}\cdots k_{r}}$can take integer values for at most finite many integers $n$. In 2012, Chenand Tang refined this result by showing that $H_{n}(\{1\}^r)$ is an integeronly for $(n,r)=(1,1)$ and $(n,r)=(3,2)$. In this paper, we consider theintegrality problem for the first and second elementary symmetric functionof $1, 1/2^{s_2}, ...,$ $1/n^{s_n}$, we show that none of themis an integer with some natural exceptions.

ON SOME MULTIPLE CHARACTER SUMS

We improve a recent result of B. Hanson (2015) on multiplicative character sums with expressions of the type $a + b +cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a + b(c+d)$. These bounds rely on some recent advances in additive combinatorics.

The upper bounds for multiplicative sum Zagreb index of some graph operations

The upper bounds for multiplicative sum Zagreb index of some graph operations

A New Multiplicative Arithmetic-Geometric Index

In this paper, we propose a new topological index: first multiplicative arithmetic geometric index of a molecular graph. A topological index is a numeric quantity from the structural graph of a molecule. In this paper, we compute multiplicative sum connectivity index, multiplicative product connectivity index, multiplicative atom bond connectivity index, multiplicative geometric-arithmetic index for titania nanotubes. Also we compute the multiplicative arithmetic-geometric index for titania nanotubes.

The upper bounds for multiplicative sum Zagreb index of some graph operations

The upper bounds for multiplicative sum Zagreb index of some graph operations

On relations between inverse sum indeg index and multiplicative sum Zagreb index

On relations between inverse sum indeg index and multiplicative sum Zagreb index

Transformation of generalized multiple Riemann zeta type sums with repeated arguments

The aim of this paper is the study of a transformation dealing with the general K-fold infinite series of the form∑n1≥⋯≥nK≥1∏j=1Kanj, especially those, where an=R(n) is a rational function satisfying certain simple conditions. These sums represent the direct generalization of the well-known multiple Riemann zeta-star function with repeated arguments ζ⋆({s}K) when an=1/ns. Our result reduces ∑∏anj to a special kind of one-fold infinite series. We apply the main theorem to the rational function R(n)=1/((n+a)s+bs) in case of which the resulting K-fold sum is called the generalized multiple Hurwitz zeta-star function ζ⋆(a,b;{s}K). We construct an effective algorithm enabling the complete evaluation of ζ⋆(a,b;{2s}K) with a∈{0,−1/2}, b∈R∖{0}, (K,s)∈N2, by means of a differential operator and present a simple ‘Mathematica’ code that allows their symbolic calculation. We also provide a new transformation of the ordinary multiple Riemann zeta-star values ζ⋆({2s}K) and ζ⋆({3}K) corresponding to a=b=0.

A family of super congruences involving multiple harmonic sums

In recent years, the congruence [Formula: see text] first discovered by the last author has been generalized by either increasing the number of indices and considering the corresponding super congruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar super congruences modulo prime powers [Formula: see text] with the indices summing up to [Formula: see text] where [Formula: see text] is coprime to [Formula: see text], and where all the indices are also coprime to [Formula: see text].

A note on moments of limit log-infinitely divisible stochastic measures of Bacry and Muzy

A multiple integral representation of single and joint moments of the total mass of the limit log-infinitely divisible stochastic measure of Bacry and Muzy [$\textit{Comm. Math. Phys.}$ ${\bf 236}$: 449-475, 2003] is derived. The covariance structure of the total mass of the measure is shown to be logarithmic. A generalization of the Selberg integral corresponding to single moments of the limit measure is proposed and shown to satisfy a recurrence relation. The joint moments of the limit lognormal measure, classical Selberg integral with $\lambda_1=\lambda_2=0,$ and Morris integral are represented in the form of multiple binomial sums. For application, low moments of the limit log-Poisson measure are computed exactly and low joint moments of the limit lognormal measure are considered in detail.

Identity families of multiple harmonic sums and multiple zeta star values

In this paper we present many new families of identities for multiple harmonic sums using binomial coefficients. Some of these generalize a few recent results of Hessami Pilehrood et al. As applications we prove several conjectures involving multiple zeta star values (MZSV): the Two-one formula conjectured by Ohno and Zudilin, and a few conjectures of Imatomi et al. involving 2-3-2-1 type MZSV, where 2 means some finite string of 2's

Estimation of the offspring mean in a branching process with non stationary immigration

ABSTRACTIn the paper, we consider a natural estimator of the offspring mean of a branching process with non stationary immigration based on observation of population sizes and number of immigrating individuals to each generation. We demonstrate that using a central limit theorem for multiple sums of dependent random variables it is possible to derive asymptotic distributions for the estimator without prior knowledge about the behavior (criticality) of the reproduction process. Before the three cases of criticality have been considered separately. Assuming that the immigration mean and variance vary regularly, conditions guaranteeing the strong consistency of the proposed estimator is also derived.

Multiplication formulas for q-Appell polynomials and the multiple q-power sums

In the first article on q-analogues of two Appell polynomials, the generalized Apostol-Bernoulli and Apostol-Euler polynomials, focus was on generalizations, symmetries, and complementary argument theorems. In this second article, we focus on a recent paper by Luo, and one paper on power sums by Wang and Wang. Most of the proofs are made by using generating functions, and the (multiple) q-addition plays a fundamental role. The introduction of the q-rational numbers in formulas with q-additions enables natural q-extension of vector forms of Raabes multiplication formulas. As special cases, new formulas for q-Bernoulli and q-Euler polynomials are obtained.

Decomposition of (Co)isotropic Relations

We identify thirteen isomorphism classes of indecomposable coisotropic relations between Poisson vector spaces and show that every coisotropic relation between finite-dimensional Poisson vector spaces may be decomposed as a direct sum of multiples of these indecomposables. We also find a list of thirteen invariants, each of which is the dimension of a space constructed from the relation, such that the 13-vector of multiplicities and the 13-vector of invariants are related by an invertible matrix over $\mathbb Z$. It turns out to be simpler to do the analysis above for isotropic relations between presymplectic vector spaces. The coisotropic/Poisson case then follows by a simple duality argument.