A generalization of Ohnoʼs relation for multiple zeta values

In this paper, we prove that certain parametrized multiple series satisfy the same relation as Ohno?s relation for multiple zeta values. This result gives us a generalization of Ohno?s relation for multiple zeta values. By virtue of this generalization, we obtain a certain equivalence between the above relation among the parametrized multiple series and a subfamily of the relation. As applications of the above results, we obtain some results on multiple zeta values.     

Integrals over polytopes, multiple zeta values and polylogarithms, and Euler’s constant

Let T be the triangle with vertices (1, 0), (0, 1), (1, 1). We study certain integrals over T, one of which was computed by Euler. We give expressions for them both as linear combinations of multiple zeta values, and as polynomials in single zeta values. We obtain asymptotic expansions of the integrals, and of sums of certain multiple zeta values with constant weight. We also give related expressions for Euler’s constant, and study integrals, one of which is the iterated Chen (Drinfeld-Kontsevich) integral, over some polytopes that are higher-dimensional analogs of T. The latter leads to a relation between certain multiple polylogarithm values and multiple zeta values.     

On the duality and the derivation relations for multiple zeta values

We consider the problem of deducing the duality relation from the extended double shuffle relation for multiple zeta values. Especially we prove that the duality relation for double zeta values and that for the sum of multiple zeta values whose first components are 2’s are deduced from the derivation relation, which is known as a subclass of the extended double shuffle relation.     

Conical zeta values and their double subdivision relations

We introduce the concept of a conical zeta value as a geometric generalization of a multiple zeta value in the context of convex cones. The quasi-shuffle and shuffle relations of multiple zeta values are generalized to open cone subdivision and closed cone subdivision relations respectively for conical zeta values. In order to achieve the closed cone subdivision relation, we also interpret linear relations among fractions as subdivisions of decorated closed cones. As a generalization of the double shuffle relation of multiple zeta values, we give the double subdivision relation of conical zeta values and formulate the extended double subdivision relation conjecture for conical zeta values.     

Rotation number values at a discontinuity

In the space of orientation-preserving circle maps that are not necessarily surjective nor injective, the rotation number does not vary continuously. Each map where one of these discontinuities occurs is itself discontinuous and we can consider the possible values of the rotation number when we modify this map only at its discontinuities. These values are always rational numbers that necessarily obey a certain arithmetic relation. In this paper we show that in several examples this relation totally characterizes the possible values of the rotation number on its discontinuities, but we also prove that in certain circumstances this relation is not sufficient for this characterization.     

求解全局最优问题的多重点样本水平值估计的相对熵算法

研究有界闭箱约束下的全局最优化问题,利用相对熵及广义方差函数方程的最大根与全局最小值之间的等价关系,设计求解全局最优值的积分型水平值估计算法.对采用重点样本采样技巧产生的函数值按一定规则进行聚类,从而在各聚类中产生的若干新重点样本,结合相对熵算法,构造出多重点样本进行全局搜索的新算法.该算法的优点在于每次迭代选用当前较好的函数值信息,以达到随机搜索到更好的函数值信息.同时多重点样本可有利挖掘出更好的全局信息.一系列的数值实验表明该算法是非常有效的.     

A NEW METHOD OF PRODUCING FUNCTIONAL RELATIONS AMONG MULTIPLE ZETA-FUNCTIONS

In this paper, we introduce a new method of producing functionalrelations among multiple zeta-functions. This method can beregarded as a kind of multiple analogue of Hardy's one of provingthe functional equation for the Riemann zeta-function. Usingthis method, we give new functional relations for multiple zeta-functions.In particular, substituting positive integers into variablesof them, we obtain known relation formulas for the multiplezeta-values. Furthermore, applying our method to a certain seriesinvolving hyperbolic sine functions, we can obtain certain multipleanalogues of the known results given by Cauchy, Ramanujan, Berndtand so on.     

The Chowla-Selberg formula

The Chowla-Selberg formula is a monomial relation connecting the values of certain automorphic form at special points to the values of Γ functions at rational points. A generalization of this formula is established in the context of CM-fields: the values of a distinguished Hilbert automorphic form at special points are expressed in terms of multiple Γ functions.     

Generalized Jacobi–Trudi determinants and evaluations of Schur multiple zeta values

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi–Trudi formulas and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulas we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.     

Quadratic relations for a q-analogue of multiple zeta values

We obtain a class of quadratic relations for a q-analogue of multiple zeta values (qMZV’s). In the limit q→1, it turns into Kawashima’s relation for multiple zeta values. As a corollary we find that qMZV’s satisfy the linear relation contained in Kawashima’s relation. In the proof we make use of a q-analogue of Newton series and Bradley’s duality formula for finite multiple harmonic q-series.     

On multiple series of Eisenstein type

The aim of this paper is to study certain multiple series which can be regarded as multiple analogues of Eisenstein series. As part of a prior research, the second-named author considered double analogues of Eisenstein series and expressed them as polynomials in terms of ordinary Eisenstein series. This fact was derived from the analytic observation of infinite series involving hyperbolic functions which were based on the study of Cauchy, and also Ramanujan. In this paper, we prove an explicit relation formula among these series. This gives an alternative proof of this fact by using the technique of partial fraction decompositions of multiple series which was introduced by Gangl, Kaneko and Zagier. By the same method, we further show a certain multiple analogue of this fact and give some examples of explicit formulas. Finally we give several remarks about the relation between the results of the present and the previous works for infinite series involving hyperbolic functions.     

A class of relations among multiple zeta values

We prove a new class of relations among multiple zeta values (MZV's) which contains Ohno's relation. We also give the formula for the maximal number of independent MZV's of fixed weight, under our new relations. To derive our formula for MZV's, we consider the Newton series whose values at non-negative integers are finite multiple harmonic sums.     

The Algebra and Combinatorics of Shuffles andMultiple Zeta Values

The algebraic and combinatorial theory of shuffles, introduced by Chen and Ree, is further developed and applied to the study of multiple zeta values. In particular, we establish evaluations for certain sums of cyclically generated multiple zeta values. The boundary case of our result reduces to a former conjecture of Zagier.     

On some functional relations between Mordell-Tornheim double L-functions and Dirichlet L-functions

In the past decade, many relation formulas for the multiple zeta values, further for the multiple L-values at positive integers have been discovered. Recently Matsumoto suggested that it is important to reveal whether those relations are valid only at integer points, or valid also at other values. Indeed the famous Euler formula for ζ(2k) can be regarded as a part of the functional equation of ζ(s). In this paper, we give certain analytic functional relations between the Mordell-Tornheim double L-functions and the Dirichlet L-functions of conductor 3 and 4. These can be regarded as continuous generalizations of the known discrete relations between the Mordell-Tornheim L-values and the Dirichlet L-values of conductor 3 and 4 at positive integers.     

Multiple harmonic series related to multiple Euler numbers

In this paper, we define the multiple Euler numbers and consider some multiple harmonic series of Mordell-Tornheim's type, which is a partial sum of the Mordell-Tornheim zeta series defined by Matsumoto. Indeed, we prove a certain reducibility of these series as well as the multiple zeta values.     

An algebraic proof of the cyclic sum formula for multiple zeta values

We introduce an algebraic formulation of the cyclic sum formulas for multiple zeta values and for multiple zeta-star values. We also present an algebraic proof of cyclic sum formulas for multiple zeta values and for multiple zeta-star values by reducing them to Kawashima's relation.     

The deficiencies of meromorphic solutions of certain algebraic differential equations

We use the spread relation to prove estimates that contain the Nevanlinna deficiencies of values of meromorphic solutions of certain differential equations of the form (1.1) below. We construct examples which show that all of our estimates are sharp, and in most of these constructions we use functions which are extremal for the spread relation. Several other examples are also given to illustrate our results.     

Asymptotics of resolvent integrals: The suppression of crossings for analytic lattice dispersion relations

We study the so-called crossing estimate for analytic dispersion relations of periodic lattice systems in dimensions three and higher. Under a certain regularity assumption on the behaviour of the dispersion relation near its critical values, we prove that an analytic dispersion relation suppresses crossings if and only if it is not a constant on any affine hyperplane. In particular, this applies to any dispersion relation which is an analytic Morse function. We also provide two examples of simple lattice systems whose dispersion relations do not suppress crossings in the present sense.     

A note on sums of products of Bernoulli numbers

In this work we obtain a new approach to closed expressions for sums of products of Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function.     

Equivariant holomorphic differential operators and finite averages of values of L-functions

Using pullback formulas for both Siegel-Eisenstein series and Jacobi-Eisenstein series the second author obtained relations between critical values of certain L-functions. To extend these relations to other critical values we use holomorphic differential operators for both types of pullbacks. The differential operators in question are well known in the Siegel case whereas for the Jacobi case they have to be developed from scratch. To compare the two pullbacks, we have furthermore to establish a relation of unexpected nature between the two types of differential operators.