Weighted Sum Formulas Research Articles

Weighted sum formula for variants of half multiple zeta values

We prove some weighted sum formulas for half multiple zeta values, half finite multiple zeta values, and half symmetric multiple zeta values. The key point of our proof is Dougall’s identity for the generalized hypergeometric function [Formula: see text]. Similar results for interpolated refined symmetric multiple zeta values and half refined symmetric multiple zeta values are also discussed.

Relations of multiple t-values of general level

We study the relations of multiple [Formula: see text]-values of general level. The generating function of sums of multiple [Formula: see text]-(star) values of level [Formula: see text] with fixed weight, depth and height is represented by the generalized hypergeometric function [Formula: see text], which generalizes the results for multiple zeta(-star) values and multiple [Formula: see text]-(star) values. As applications, we obtain formulas for the generating functions of sums of multiple [Formula: see text]-(star) values of level [Formula: see text] with height one and maximal height and a weighted sum formula for sums of multiple [Formula: see text]-(star) values of level [Formula: see text] with fixed weight and depth. Using the stuffle algebra, we also get the symmetric sum formulas and Hoffman’s restricted sum formulas for multiple [Formula: see text]-(star) values of level [Formula: see text]. Some evaluations of multiple [Formula: see text]-star values of level [Formula: see text] with one–two–three indices are given.

Further results on pinnacle sets

The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis, Nelson, Petersen, and Tenner and also results proven recently by Fang, this paper aims to add to our understanding of pinnacle sets by giving a simpler and more combinatorial proof of a weighted sum formula previously proven by Fang.

Weighted Sum Formulas from Shuffle Products of Multiple Zeta-Star Values

In this paper, we are going to perform the shuffle products of $$Z_-({m}) = \sum _{a+b=m} (-1)^{b} \zeta (\{1\}^{a},b+2)$$ and $$Z_+^\star (n) = \sum _{c+d=n} \zeta ^{\star }(\{1\}^{c},d+2)$$ with $$m+n = p$$ . The resulted shuffle relation is a weighted sum formula stated below: $$\begin{aligned}&\frac{(p+1)(p+2)}{2} \zeta (p+4) =\sum _{\begin{array}{c} m+n=p\\ \mid \varvec{\alpha }\mid =p+3 \end{array}} \zeta (\alpha _{0}, \alpha _{1}, \ldots , \alpha _{m}, \alpha _{m+1}+1) \\&\times {\sum _{a+b+c=m}}^{*} \Bigl ( W_{\varvec{\alpha }}(a,b,c) + W_{\varvec{\alpha }}(a,b,0) + W_{\varvec{\alpha }}(0,b,c)+ W_{\varvec{\alpha }}(0,m,0) \Bigr ), \end{aligned}$$ where the summation $$\sum ^*$$ and the weights $$W_{\varvec{\alpha }}(a,b,c)$$ are given appropriately. We also give some weighted alternating Euler sums formulas.

OHNO–ZAGIER TYPE RELATIONS FOR MULTIPLE t-VALUES

AbstractWe study Ohno–Zagier type relations for multiple t-values and multiple t-star values. We represent the generating function of sums of multiple t-(star) values with fixed weight, depth and height in terms of the generalised hypergeometric function $\,_3F_2$ . As applications, we get a formula for the generating function of sums of multiple t-(star) values of maximal height and a weighted sum formula for sums of multiple t-(star) values with fixed weight and depth.

Proof of Kaneko–Tsumura conjecture on triple T-values

Many [Formula: see text]-linear relations exist between multiple zeta values, the most interesting of which are the weighted sum formulas. In this paper, we generalize these to Euler sums and Kaneko and Tsumura’s multiple [Formula: see text]-values, a variant of the multiple zeta values, by considering generating functions of the Euler sums. Through this approach, we are able to re-prove a few known formulas, confirm a conjecture of Kaneko and Tsumura on triple [Formula: see text]-values, and discover many new identities.

Weighted multiple blockwise imputation method for high-dimensional regression with blockwise missing data

Blockwise missing data present a great challenge for data analysis because of their special missing structure. In this article, we propose a novel weighted multiple blockwise imputation method to target the problem of high-dimensional regression with blockwise missing data. Specifically, we first apply a multiple blockwise imputation technique to impute the missing blocks in the design matrix, after which a weighted sum formula is used to integrate the resulting imputation schemes and estimate the regression coefficients. The proposed method can make full use of the available information to perform imputation; hence, it shows superior performance compared to some previous methods. Simulations illustrate the advantageous numerical performance of our method in terms of variable selection, parameter estimation, and prediction ability. A real application is also presented to demonstrate its practical merit.

Algebraic relations of interpolated multiple zeta values

Interpolated multiple zeta values can be regarded as interpolation polynomials of multiple zeta values and multiple zeta-star values. In this paper, we give some algebraic relations of interpolated multiple zeta values, such as the shuffle regularized sum formula, a weighted sum formula and some evaluation formulas with even arguments. All the algebraic relations provided in this paper are deduced from the extended double shuffle relations.

Yamamoto's Interpolation of Finite Multiple Zeta and Zeta-star Values

We study a polynomial interpolation of finite multiple zeta and zeta-star values with variable $t$, which is an analogue of interpolated multiple zeta values introduced by Yamamoto. We introduce several relations among them and, in particular, prove the cyclic sum formula, the Bowman--Bradley type formula, and the weighted sum formula. The harmonic relation, the shuffle relation, the duality relation, and the derivation relation are also presented.

Some relations deduced from regularized double shuffle relations of multiple zeta values

It is conjectured that the regularized double shuffle relations give all algebraic relations among the multiple zeta values, and hence all other algebraic relations should be deduced from the regularized double shuffle relations. In this paper, we provide as many as the relations which can be derived from the regularized double shuffle relations, for example, the weighted sum formula of Guo and Xie, some evaluation formulas with even arguments and the restricted sum formulas of Hoffman and their generalizations.

Weighted sum formulas of multiple t-values with even arguments

Abstract In this paper, we study the weighted sums of multiple t-values and of multiple t-star values at even arguments. Some general weighted sum formulas are given, where the weight coefficients are given by (symmetric) polynomials of the arguments.

On sum formulas for Mordell–Tornheim zeta values

In this paper we prove new sum formulas for Mordell–Tornheim zeta values in the case of depth 2 and 3, expressing the sums as single multiples of Riemann zeta values. Also, we obtain weighted sum formulas for double Mordell–Tornheim zeta values. Moreover, we present a sum formula for the Mordell–Tornheim series of even arguments.

On some weighted sum formulas involving general multiple zeta-type series

The purpose of this paper is a study of the general finite sumsΦN,d(K):=∑N≥n1≥⋯≥nK≥1∏j=1KA⌈njd⌉,d∈N, that generalize the truncated multiple harmonic sums ζN⋆({s}K) corresponding to d=1 and An=1/ns with s∈N. Surprisingly, when specializing our general transformation result concerning ΦdN,d(K), such a type of finite sums can be used for generating and closed-form evaluation of new linear combinations of multiple Hurwitz zeta-star values of the form∑s⊢Kmax⁡(s)≤dζ⋆(cs;a)⋅∏r=1ℓ(s)((−1)sr−1⋅(dsr)), assuming (a,c,d,K)∈R×N3 with a>−1, c>1, where the sum is extended over all compositions of K with maximal part not exceeding d.

Weighted sum formula for multiple harmonic sums modulo primes

In this paper we prove a weighted sum formula for multiple harmonic sums modulo primes, thereby proving a weighted sum formula for finite multiple zeta values. Our proof utilizes difference equations for the generating series of multiple harmonic sums. We also conjecture several weighted sum formulas of similar flavor for finite multiple zeta values.

Weighted sum formulas of multiple zeta values with even arguments

We obtain a weighted sum formula of the zeta values at even arguments, and a weighted sum formula of the multiple zeta values with even arguments and its zeta-star analogue. The weight coefficients are given by (symmetric) polynomials of the arguments. These weighted sum formulas for the zeta values and for the multiple zeta values were conjectured by L. Guo, P. Lei and J. Zhao.

Weighted sum formulas for finite multiple zeta values

We prove a formula among finite multiple zeta values with four parameters. This formula is proved by using the iterated integral expression of the multiple polylogarithms. By specializing these parameters, we give some weighted sum formulas for finite multiple zeta values.

A Utilization-based Genetic Algorithm for Solving the University Timetabling Problem (UGA)

Building university timetables is a complex process that considers varying types of constraints and objectives from one institution to another. The problem solved in this paper is a real one featuring a number of hard and soft constraints that are not very conventional. The pursued objective is also novel and considers maximizing resource utilization. This paper introduces a genetic algorithm that uses some heuristics to generate an initial population of feasible good quality timetables. The algorithm uses a simple weighted sum formula to respect professors’ preferences and handle conflicts. In order to reduce waste, a crossover type focusing on the utilization rates of learning spaces is introduced. A targeted mutation operator that uses a local search heuristic is also employed. The algorithm applies a composite fitness function that considers space utilization, gaps between events and a maximum number of lectures per day. A large dataset with real data from the Faculty of Commerce, Alexandria University in Egypt was used to test the contributed algorithm. The algorithm was also tested against two difficult benchmark problems from the literature. Testing proved that the developed algorithm is an effective tool for managing timetables and resources in universities. It performed remarkedly well on the large datasets of the two benchmark problems and it also respected more constraints than those stated in the initial problem statement of the two benchmark datasets.

On Vectorized Weighted Sum Formulas of Multiple Zeta Values

In this paper, we introduce the vectorization of shuffle products of two sums of multiple zeta values, which generalizes the weighted sum formula obtained by Ohno and Zudilin. Some interesting weighted sum formulas of vectorized type are obtained, such as\begin{align*} &\quad \sum_{\substack{\boldsymbol{\alpha}+\boldsymbol{\beta}=\boldsymbol{k} |\boldsymbol{\alpha}|: \textrm{even}}} M(\boldsymbol{\alpha}) M(\boldsymbol{\beta}) \sum_{|\boldsymbol{c}|=|\boldsymbol{k}|+r+3} 2^{c_{|\boldsymbol{\alpha}|+1}} \zeta(c_0, c_1, \ldots, c_{|\boldsymbol{\alpha}|}, \ldots, c_{|\boldsymbol{k}|+1}+1) \\ &= \frac{1}{2} (2|\boldsymbol{k}|+r+5) M(\boldsymbol{k}) \zeta(|\boldsymbol{k}|+r+4),\end{align*}where $\boldsymbol{\alpha}$, $\boldsymbol{\beta}$ and $\boldsymbol{k}$ are $n$-tuples of nonnegative integers with $|\boldsymbol{k}| = k_1 + k_2 + \cdots + k_n$ even; $M(\boldsymbol{u})$ is the multinomial coefficient defined by $\binom{u_1 + u_2 + \cdots + u_n}{u_1, u_2, \ldots, u_n}$ with the value $\frac{|\boldsymbol{u}|!}{u_1! u_2! \cdots u_n!}$; and $r$ is a nonnegative integer. Moreover, some newly developed combinatorial identities of vectorized types involving multinomial coefficients by extending the shuffle products of two sums of multiple zeta values in their vectorizations are given as well.

Families of weighted sum formulas for multiple zeta values

Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper, we prove a family of identities involving Bernoulli numbers and apply them to obtain infinitely many weighted sum formulas for double zeta values and triple zeta values where the weight coefficients are given by symmetric polynomials. We give a general conjecture in arbitrary depth at the end of the paper.

Weighted sum formulas from shuffle products of multiples of Riemann zeta values

The classical Euler decomposition theorem expressed a product of two Riemann zeta values in terms of double Euler sums. It can also be obtained from the shuffle product of two Riemann zeta values when both are expressed as iterated integrals due to Kontsevich. In this paper, we investigate the shuffle product of n multiples of Riemann zeta values(cj+1)ζ(cj+2),j=1,2,…,n,cj≥0 through their particular integral representations. Among other things, we obtain the following weighted sum formula∑|α|=k∑βζ(α1+β1+1,α2+β2+1,…,αn+βn+1)∏j=1n(αj+βj)!αj!jαj=1n!∑|c|=k∏j=1n(cj+1)ζ(cj+2) with∑βx1β1x2β2⋯xnβn=(x1+x2+⋯+xn)(x2+⋯+xn)⋯xn which is a sum of n! monomials of degree n. Here ζ(a1,a2,…,an) appearing in the summation is a multiple zeta value or n-fold Euler sum. Also we obtain the weighted sum formula of triple Euler sums∑|α|=k+5ζ(α1,α2,α3+1)⋅{2α2−1(3α3−2α3)−(2α3−1)}−∑|α|=k+4ζ(1,α2,α3+1)⋅(2α3−1)+ζ(1,1,k+4)=16∑|c|=kζ(c1+2)ζ(c2+2)ζ(c3+2).