Mathematische Zeitschrift, ISSN 0025-5874, 4/2019, Volume 291, Issue 3, pp. 1337 - 1356

We prove 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...

Weighted sum formulas | Bernoulli numbers | 11M32 | Multiple zeta values | Mathematics, general | Mathematics | Multiple zeta-star values | 11B68 | MATHEMATICS

Weighted sum formulas | Bernoulli numbers | 11M32 | Multiple zeta values | Mathematics, general | Mathematics | Multiple zeta-star values | 11B68 | MATHEMATICS

Journal Article

Taiwanese journal of mathematics, ISSN 1027-5487, 4/2016, Volume 20, Issue 2, pp. 243 - 261

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...

Integers | Mathematics | Weighted sum formula | Multiple zeta value | Shuffle relation | Sum formula | MATHEMATICS

Integers | Mathematics | Weighted sum formula | Multiple zeta value | Shuffle relation | Sum formula | MATHEMATICS

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 08/2013, Volume 9, Issue 5, pp. 1185 - 1198

...) or sums of multiple zeta values of fixed weight and depth. Some interesting weighted sum formulas are obtained, such as $\sum_{\vert {\boldsymbol{\alpha}} \vert = m...

weighted sum formula | shuffle relation | sum formula | Multiple zeta value | MATHEMATICS | HARMONIC SERIES

weighted sum formula | shuffle relation | sum formula | Multiple zeta value | MATHEMATICS | HARMONIC SERIES

Journal Article

Proceedings of the American Mathematical Society, ISSN 0002-9939, 09/2017, Volume 145, Issue 9, pp. 3795 - 3808

Let r(k)(n) denote the number of representations of the positive integer n as the sum of k squares...

Dirichlet characters | Dirichlet series | Sums of squares | Ramanujan’s lost notebook | Bessel functions | Voronoï summation formula | MATHEMATICS | Ramanujan's lost notebook | BESSEL-FUNCTION SERIES | WEIGHTED DIVISOR SUMS | MATHEMATICS, APPLIED | Dirichlet-series | IDENTITIES | Voronoi summation formula

Dirichlet characters | Dirichlet series | Sums of squares | Ramanujan’s lost notebook | Bessel functions | Voronoï summation formula | MATHEMATICS | Ramanujan's lost notebook | BESSEL-FUNCTION SERIES | WEIGHTED DIVISOR SUMS | MATHEMATICS, APPLIED | Dirichlet-series | IDENTITIES | Voronoi summation formula

Journal Article

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS, ISSN 0035-7596, 02/2020, Volume 50, Issue 1, pp. 225 - 235

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...

MATHEMATICS | sum formulas | Mordell-Tornheim zeta values | weighted sum formulas | multiple zeta values

MATHEMATICS | sum formulas | Mordell-Tornheim zeta values | weighted sum formulas | multiple zeta values

Journal Article

Frontiers of mathematics in China, ISSN 1673-3576, 2017, Volume 12, Issue 5, pp. 1183 - 1200

... > 0, consider the sum S X (f; α, β) = ∑ n λ f (n)e(αn β )ϕ(n/X), where ϕ is a smooth function of compact support...

resonance sum | Maass form | first derivative test | weighted stationary phase | Cusp form | Fourier coefficient of cusp form | Kuznetsov trace formula | Mathematics, general | Mathematics | 11L07 | 11F30 | EXPONENTIAL-SUMS | GL | MATHEMATICS | BOUNDS | COEFFICIENTS | SELBERG L-FUNCTIONS | CUSP FORMS | SUBCONVEXITY | Fourier analysis | Asymptotic series | Laplace transforms | Infinity | Mathematical analysis | Eigen values

resonance sum | Maass form | first derivative test | weighted stationary phase | Cusp form | Fourier coefficient of cusp form | Kuznetsov trace formula | Mathematics, general | Mathematics | 11L07 | 11F30 | EXPONENTIAL-SUMS | GL | MATHEMATICS | BOUNDS | COEFFICIENTS | SELBERG L-FUNCTIONS | CUSP FORMS | SUBCONVEXITY | Fourier analysis | Asymptotic series | Laplace transforms | Infinity | Mathematical analysis | Eigen values

Journal Article

Forum mathematicum, ISSN 0933-7741, 07/2020, Volume 32, Issue 4, pp. 965 - 976

.... Some general weighted sum formulas are given, where the weight coefficients are given by (symmetric) polynomials of the arguments.

multiple zeta-star values | Bernoulli numbers | weighted sum formulas | 11M32 | 11B68 | multiple zeta values | MATHEMATICS | MATHEMATICS, APPLIED | IDENTITIES | ZETA VALUES | Multiple t-values | multiple t-star values | Polynomials

multiple zeta-star values | Bernoulli numbers | weighted sum formulas | 11M32 | 11B68 | multiple zeta values | MATHEMATICS | MATHEMATICS, APPLIED | IDENTITIES | ZETA VALUES | Multiple t-values | multiple t-star values | Polynomials

Journal Article

Journal of Approximation Theory, ISSN 0021-9045, 09/2015, Volume 197, pp. 101 - 114

Let r2(n) denote the number of representations of n as a sum of two squares. Finding the precise order of magnitude for the error term in the asymptotic formula for ∑n≤xr2(n...

Riesz sums | Weighted divisor sums | Ramanujan’s lost notebook | Bessel functions | Dirichlet divisor problem | Dirichlet [formula omitted]-series | Ramanujan's lost notebook | Dirichlet L-series | MATHEMATICS

Riesz sums | Weighted divisor sums | Ramanujan’s lost notebook | Bessel functions | Dirichlet divisor problem | Dirichlet [formula omitted]-series | Ramanujan's lost notebook | Dirichlet L-series | MATHEMATICS

Journal Article

数学学报：英文版, ISSN 1439-8516, 2016, Volume 32, Issue 7, pp. 797 - 806

We obtain the weighted sum identities for.

恒等式 | 加权求和 | zeta值 | 11M41 | Double zeta values | geometric sum | Mathematics, general | Mathematics | 11M06 | weighted | MATHEMATICS | MATHEMATICS, APPLIED | Studies | Theorems | Mathematical models | Texts | Formulas (mathematics)

恒等式 | 加权求和 | zeta值 | 11M41 | Double zeta values | geometric sum | Mathematics, general | Mathematics | 11M06 | weighted | MATHEMATICS | MATHEMATICS, APPLIED | Studies | Theorems | Mathematical models | Texts | Formulas (mathematics)

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 12/2012, Volume 8, Issue 8, pp. 1903 - 1921

A typical formula of multiple zeta values is the sum formula which expresses a Riemann zeta value as a sum of all multiple zeta values of fixed weight and depth...

Parametrized sum formula | Multiple polylogarithm | Weighted sum formula | Multiple zeta value | weighted sum formula | MIXED TATE MOTIVES | MATHEMATICS | multiple polylogarithm | HARMONIC SERIES | DUALITY | HEIGHT | parametrized sum formula | STAR VALUES | Mathematics - Number Theory

Parametrized sum formula | Multiple polylogarithm | Weighted sum formula | Multiple zeta value | weighted sum formula | MIXED TATE MOTIVES | MATHEMATICS | multiple polylogarithm | HARMONIC SERIES | DUALITY | HEIGHT | parametrized sum formula | STAR VALUES | Mathematics - Number Theory

Journal Article

Journal of Number Theory, ISSN 0022-314X, 07/2018, Volume 188, pp. 247 - 262

In this paper, we are going to evaluate a family {Ep(2m,n,k)|p∈Z} of sums of multiple zeta values with even arguments and polynomial weights defined asEp(2m,n,k)=∑α1+⋯+αk=n∏j=1k(αj+p−1αj)×ζ(2mα1,2mα2,⋯,2mαk...

Multiple zeta values | Weighted sum formula | Multiple zeta-star value | MATHEMATICS

Multiple zeta values | Weighted sum formula | Multiple zeta-star value | MATHEMATICS

Journal Article

Kyushu journal of mathematics, ISSN 1340-6116, 2017, Volume 71, Issue 1, pp. 197 - 209

... of Dirichlet L-function values for short interval character sums and in this sense our treatment is decisive, i.e...

cosine higher-order Euler numbers | generalized Bernoulli number | Dirichlet class number formula | Dirichlet L-function | weighted short-interval character sum | Cosine higher-order Euler numbers | Weighted short-interval character sum | Generalized Bernoulli number | MATHEMATICS | BERNOULLI NUMBERS | SUMS

cosine higher-order Euler numbers | generalized Bernoulli number | Dirichlet class number formula | Dirichlet L-function | weighted short-interval character sum | Cosine higher-order Euler numbers | Weighted short-interval character sum | Generalized Bernoulli number | MATHEMATICS | BERNOULLI NUMBERS | SUMS

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 06/2019, Volume 15, Issue 5, pp. 1051 - 1057

Let G be a finite additive abelian group with exponent n and S = g 1 ⋯ g t be a sequence of elements in G . For any element g of G and A ⊆ { 1 , 2 , … , n − 1...

Davenport constant | Finite abelian group | weighted subsequences | Mathematics - Number Theory

Davenport constant | Finite abelian group | weighted subsequences | Mathematics - Number Theory

Journal Article

Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, ISSN 0025-5858, 4/2015, Volume 85, Issue 1, pp. 23 - 41

... or $$r$$ r -fold Euler sum $$\zeta (\alpha _{1}, \alpha _{2}, \ldots , \alpha _{r})$$ ζ ( α 1 , α 2 , … , α r ) is defined by the multiple series...

33E99 | Weighted sum formula | Multiple zeta values | Mathematics | Topology | Geometry | Algebra | Secondary 11M99 | 40B05 | Mathematics, general | Number Theory | Differential Geometry | Drinfeld integral | Shuffle product | Primary 40A25 | MATHEMATICS

33E99 | Weighted sum formula | Multiple zeta values | Mathematics | Topology | Geometry | Algebra | Secondary 11M99 | 40B05 | Mathematics, general | Number Theory | Differential Geometry | Drinfeld integral | Shuffle product | Primary 40A25 | MATHEMATICS

Journal Article

Discrete Mathematics, ISSN 0012-365X, 2010, Volume 310, Issue 21, pp. 2801 - 2805

Let G be a finite abelian group and let k ⩾ 2 be an integer. A sequence of k elements a 1 , a 2 , … , a k in G is called a k -barycentric sequence if there...

[formula omitted]-barycentric Davenport constant | [formula omitted]-barycentric sequences | K-barycentric Davenport constant | K-barycentric sequences | WEIGHTED SUMS | MATHEMATICS | k-barycentric sequences | ZERO-SUM PROBLEMS | k-barycentric Davenport constant | Integers | Mathematical analysis | Upper bounds | Images

[formula omitted]-barycentric Davenport constant | [formula omitted]-barycentric sequences | K-barycentric Davenport constant | K-barycentric sequences | WEIGHTED SUMS | MATHEMATICS | k-barycentric sequences | ZERO-SUM PROBLEMS | k-barycentric Davenport constant | Integers | Mathematical analysis | Upper bounds | Images

Journal Article

Computers and Operations Research, ISSN 0305-0548, 12/2014, Volume 52, Issue Part A, pp. 135 - 146

...s optimality criterion: the total weighted completion time or the maximum lateness. The aim is to find a complete schedule minimizing a weighted sum of the two criteria...

Total weighted completion time | formula omitted | Algorithms | Maximum lateness | Deteriorating jobs | Agent scheduling | NP-hardness | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SINGLE-MACHINE | 2-AGENT | ENGINEERING, INDUSTRIAL | PROCESSING TIMES | Schedules | Deterioration | Mathematical analysis | Lateness | Scheduling | Criteria | Optimization

Total weighted completion time | formula omitted | Algorithms | Maximum lateness | Deteriorating jobs | Agent scheduling | NP-hardness | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SINGLE-MACHINE | 2-AGENT | ENGINEERING, INDUSTRIAL | PROCESSING TIMES | Schedules | Deterioration | Mathematical analysis | Lateness | Scheduling | Criteria | Optimization

Journal Article

IET Optoelectronics, ISSN 1751-8768, 10/2015, Volume 9, Issue 5, pp. 199 - 206

The symbol error rate (SER) is investigated for colour-shift keying (CSK) modulation in visible light communication system with red/green/blue LEDs. A new...

Special Issue on Optical-Wireless Communications | symbol error rate analysis | constellation distribution | TELECOMMUNICATIONS | weighted sum | light emitting diodes | ENGINEERING, ELECTRICAL & ELECTRONIC | optical modulation | SCHEME | symbol error probability | 2D plane | red-green-blue LED | signal power | colour-shift keying modulation | visible light communication system | optical communication equipment | three-dimensional signal space | noise power | numerical Gaussian formula | OPTICS | optical noise | pseudosignal-to-noise ratio | RGB light-emitting diodes | 2D space | error statistics | Error analysis | Mathematical analysis | Symbols | Modulation | Exact solutions | Mathematical models | Constellations | Three dimensional

Special Issue on Optical-Wireless Communications | symbol error rate analysis | constellation distribution | TELECOMMUNICATIONS | weighted sum | light emitting diodes | ENGINEERING, ELECTRICAL & ELECTRONIC | optical modulation | SCHEME | symbol error probability | 2D plane | red-green-blue LED | signal power | colour-shift keying modulation | visible light communication system | optical communication equipment | three-dimensional signal space | noise power | numerical Gaussian formula | OPTICS | optical noise | pseudosignal-to-noise ratio | RGB light-emitting diodes | 2D space | error statistics | Error analysis | Mathematical analysis | Symbols | Modulation | Exact solutions | Mathematical models | Constellations | Three dimensional

Journal Article

Mediterranean journal of mathematics, ISSN 1660-5446, 2017, Volume 14, Issue 3, p. 1

In the paper, the authors establish by two approaches several explicit formulas for special values of the Bell polynomials of the second kind, derive explicit formulas for the Euler numbers...

Euler polynomial | Explicit formula | Bell polynomial of the second kind | property | Euler number | special value | double sum | weighted Stirling number | MATHEMATICS, APPLIED | BERNOULLI NUMBERS | TERMS | WEIGHTED STIRLING NUMBERS | MATHEMATICS | EXPRESSIONS | 1ST

Euler polynomial | Explicit formula | Bell polynomial of the second kind | property | Euler number | special value | double sum | weighted Stirling number | MATHEMATICS, APPLIED | BERNOULLI NUMBERS | TERMS | WEIGHTED STIRLING NUMBERS | MATHEMATICS | EXPRESSIONS | 1ST

Journal Article

IEEE access, ISSN 2169-3536, 2019, Volume 7, pp. 151610 - 151617

In this paper, the non-weighted $L_{2}$ -gain control problem is addressed for a class of asynchronously switched linear systems, where the asynchronous...

Linear systems | FUZZY-SYSTEMS | Switches | Clock-dependent Lyapunov function | Math | 1998 | H-INFINITY CONTROL | asynchronous control | MathML" xmlns:xlink="http | www | ali | 1999 | Symmetric matrices | EXPONENTIAL STABILITY | niso | L-2-GAIN | non-weighted < italic xmlns:ali="http | org | xlink" xmlns:xsi="http | Process control | " xmlns:mml="http | STABILIZATION | 2001 | italic > 2-gain | COMPUTER SCIENCE, INFORMATION SYSTEMS | CONVEX CONDITIONS | TELECOMMUNICATIONS | ENGINEERING, ELECTRICAL & ELECTRONIC | Switched systems | sum of square program | schemas | L">XMLSchema-instance"> L | Lyapunov methods

Linear systems | FUZZY-SYSTEMS | Switches | Clock-dependent Lyapunov function | Math | 1998 | H-INFINITY CONTROL | asynchronous control | MathML" xmlns:xlink="http | www | ali | 1999 | Symmetric matrices | EXPONENTIAL STABILITY | niso | L-2-GAIN | non-weighted < italic xmlns:ali="http | org | xlink" xmlns:xsi="http | Process control | " xmlns:mml="http | STABILIZATION | 2001 | italic > 2-gain | COMPUTER SCIENCE, INFORMATION SYSTEMS | CONVEX CONDITIONS | TELECOMMUNICATIONS | ENGINEERING, ELECTRICAL & ELECTRONIC | Switched systems | sum of square program | schemas | L">XMLSchema-instance"> L | Lyapunov methods

Journal Article

Journal of inequalities and applications, ISSN 1029-242X, 2018, Volume 2018, Issue 1, pp. 1 - 18

... formulas of finite-time and infinite-time ruin probabilities.

Two-dimensional risk model | Dominated-variation distributions | Uniform asymptotic formulas | 26A12 | Ruin probabilities | Analysis | 60G50 | Mathematics, general | 62P05 | Mathematics | Applications of Mathematics | Stochastic returns | MATHEMATICS, APPLIED | SARMANOV FAMILY | TAIL ASYMPTOTICS | INSURANCE | RANDOMLY WEIGHTED SUMS | RANDOM-VARIABLES | MATHEMATICS | CLAIMS | Two dimensional models | Bivariate analysis | Asymptotic properties | Dependence | Research

Two-dimensional risk model | Dominated-variation distributions | Uniform asymptotic formulas | 26A12 | Ruin probabilities | Analysis | 60G50 | Mathematics, general | 62P05 | Mathematics | Applications of Mathematics | Stochastic returns | MATHEMATICS, APPLIED | SARMANOV FAMILY | TAIL ASYMPTOTICS | INSURANCE | RANDOMLY WEIGHTED SUMS | RANDOM-VARIABLES | MATHEMATICS | CLAIMS | Two dimensional models | Bivariate analysis | Asymptotic properties | Dependence | Research

Journal Article

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