2012, University lecture series, ISBN 9780821853672, Volume 59, x, 190

Book

Pacific Journal of Mathematics, ISSN 0030-8730, 2014, Volume 272, Issue 1, pp. 201 - 226

We work out some formulas for nonlinear Euler sums involving multiple zeta values. As applications of these formulas, we give new closed form sums of several...

Euler sums | Multiple zeta values | Polygamma functions | Polylogarithm functions | Nonlinear Euler sums | Landen's identities | INTEGRALS | MATHEMATICS | nonlinear Euler sums | SERIES | multiple zeta values | RIEMANN ZETA-FUNCTION | STIRLING NUMBERS | VALUES | polygamma functions | polylogarithm functions

Euler sums | Multiple zeta values | Polygamma functions | Polylogarithm functions | Nonlinear Euler sums | Landen's identities | INTEGRALS | MATHEMATICS | nonlinear Euler sums | SERIES | multiple zeta values | RIEMANN ZETA-FUNCTION | STIRLING NUMBERS | VALUES | polygamma functions | polylogarithm functions

Journal Article

Journal of Number Theory, ISSN 0022-314X, 11/2017, Volume 180, pp. 310 - 348

For any positive integers k and j, we consider some asymptotic formulas for weighted averages of the Cohen–Ramanujan sums sk(s)(j)=∑d|k,ds|jf(d)g(kd) with any...

Asymptotic results on arithmetical functions | Euler totient function | Bernoulli polynomials | Ramanujan sums | Gamma function | gcd-sum functions | Dedekind totient function | MATHEMATICS | functions | EXTENSION | VALUES | Asymptotic results on arithmetical

Asymptotic results on arithmetical functions | Euler totient function | Bernoulli polynomials | Ramanujan sums | Gamma function | gcd-sum functions | Dedekind totient function | MATHEMATICS | functions | EXTENSION | VALUES | Asymptotic results on arithmetical

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 10/2018, Volume 466, Issue 1, pp. 789 - 805

P. Flajolet and B. Salvy [15] prove the famous theorem that a nonlinear Euler sum Si1i2⋯ir,q reduces to a combination of sums of lower orders whenever the...

Euler sum | Riemann zeta function | Polylogarithm function | Harmonic number | Tornheim type series | Multiple zeta value | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | Ricmann zeta function | VALUES | Euler sure | EXPLICIT EVALUATION

Euler sum | Riemann zeta function | Polylogarithm function | Harmonic number | Tornheim type series | Multiple zeta value | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | Ricmann zeta function | VALUES | Euler sure | EXPLICIT EVALUATION

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 10/2018, Volume 466, Issue 1, pp. 1009 - 1042

In this paper we are interested in Euler-type sums with products of harmonic numbers, Stirling numbers and Bell numbers. We discuss the analytic...

Euler sums | Riemann zeta function | Multiple zeta (star) values | Stirling numbers | Harmonic numbers | Multiple harmonic (star) numbers | INTEGRALS | MATHEMATICS | MULTIPLE ZETA-VALUES | MATHEMATICS, APPLIED | SERIES

Euler sums | Riemann zeta function | Multiple zeta (star) values | Stirling numbers | Harmonic numbers | Multiple harmonic (star) numbers | INTEGRALS | MATHEMATICS | MULTIPLE ZETA-VALUES | MATHEMATICS, APPLIED | SERIES

Journal Article

Journal of Number Theory, ISSN 0022-314X, 07/2017, Volume 176, pp. 449 - 472

Let gcd(k,j) be the greatest common divisor of the integers k and j. We establish some asymptotic formulas for weighted averages of the gcd-sum functions,...

Euler totient function | Asymptotic results on arithmetical functions | gcd-sum functions | Dedekind function | Anderson–Apostol sums | Dirichlet divisor problem | MATHEMATICS | Anderson-Apostol sums

Euler totient function | Asymptotic results on arithmetical functions | gcd-sum functions | Dedekind function | Anderson–Apostol sums | Dirichlet divisor problem | MATHEMATICS | Anderson-Apostol sums

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 03/2017, Volume 447, Issue 1, pp. 419 - 434

This paper develops an approach to evaluation of Euler related sums. The approach is based on simple integral computations. By the approach, we can obtain some...

Euler sums | Riemann zeta function | Polylogarithm function | Harmonic numbers | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | Rieman zeta function

Euler sums | Riemann zeta function | Polylogarithm function | Harmonic numbers | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | Rieman zeta function

Journal Article

Journal of Number Theory, ISSN 0022-314X, 08/2016, Volume 165, pp. 84 - 108

This paper develops an approach to evaluation of Euler sums and integrals of polylogarithm functions. The approach is based on simple Cauchy product formula...

Euler sums | Riemann zeta function | Polylogarithm functions | MATHEMATICS

Euler sums | Riemann zeta function | Polylogarithm functions | MATHEMATICS

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 07/2017, Volume 451, Issue 2, pp. 954 - 975

In this paper, by using the method of Contour Integral Representations and the Theorem of Residues and integral representations of series, we discuss the...

Euler sum | Riemann zeta function | Harmonic number | Hurwitz zeta function | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED

Euler sum | Riemann zeta function | Harmonic number | Hurwitz zeta function | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 05/2017, Volume 40, Issue 7, pp. 2347 - 2361

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial...

functional equations | central factorial numbers | binomial coefficients | array polynomials | Euler numbers and polynomials | generating functions | binomial sum | Stirling numbers | combinatorial sum | MATHEMATICS, APPLIED | IDENTITIES | ARRAY TYPE POLYNOMIALS | GENERATING-FUNCTIONS | BERNOULLI | Computation | Factorials | Mathematical analysis | Mathematical models | Polynomials | Arrays | Combinatorial analysis | Sums | Mathematics - Number Theory

functional equations | central factorial numbers | binomial coefficients | array polynomials | Euler numbers and polynomials | generating functions | binomial sum | Stirling numbers | combinatorial sum | MATHEMATICS, APPLIED | IDENTITIES | ARRAY TYPE POLYNOMIALS | GENERATING-FUNCTIONS | BERNOULLI | Computation | Factorials | Mathematical analysis | Mathematical models | Polynomials | Arrays | Combinatorial analysis | Sums | Mathematics - Number Theory

Journal Article

Journal of Number Theory, ISSN 0022-314X, 08/2017, Volume 177, pp. 443 - 478

In this paper, we establish some expressions of series involving harmonic numbers and Stirling numbers of the first kind in terms of multiple zeta values, and...

Multiple star harmonic number | Euler sum | Multiple zeta star value | Multiple zeta value | Multiple harmonic number | INTEGRALS | MATHEMATICS | SERIES | NUMBERS | DUALITY | FORMULAS | RIEMANN ZETA | Mathematics - Number Theory

Multiple star harmonic number | Euler sum | Multiple zeta star value | Multiple zeta value | Multiple harmonic number | INTEGRALS | MATHEMATICS | SERIES | NUMBERS | DUALITY | FORMULAS | RIEMANN ZETA | Mathematics - Number Theory

Journal Article

Mathematical Methods in the Applied Sciences, ISSN 0170-4214, 03/2019, Volume 42, Issue 4, pp. 1082 - 1098

Recently, several works are done on the generalized Dedekind‐Vasyunin sum Capq=−qa∑k=1q−1ζ−a,kpqcotπkq, where a∈C,p and q are positive coprime integers, and...

fractional part function | Dedekind‐Vasyunin‐cotangent sum | Riemann hypothesis | Dedekind-Vasyunin-cotangent sum | MATHEMATICS, APPLIED | EULER | Integers | Reciprocity

fractional part function | Dedekind‐Vasyunin‐cotangent sum | Riemann hypothesis | Dedekind-Vasyunin-cotangent sum | MATHEMATICS, APPLIED | EULER | Integers | Reciprocity

Journal Article

Journal of Number Theory, ISSN 0022-314X, 05/2017, Volume 174, pp. 40 - 67

In this paper, we develop an approach to evaluation of nonlinear Euler sums. The approach is based on Tornheim type series computations. By the approach, we...

Euler sums | Riemann zeta function | Polylogarithm function | Tornheim type series | Harmonic numbers | INTEGRALS | MATHEMATICS

Euler sums | Riemann zeta function | Polylogarithm function | Tornheim type series | Harmonic numbers | INTEGRALS | MATHEMATICS

Journal Article

Annual Review of Physical Chemistry, ISSN 0066-426X, 4/2015, Volume 66, Issue 1, pp. 189 - 216

Sum-frequency generation vibrational spectroscopy (SFG-VS) can provide detailed information and understanding of the molecular composition, interactions, and...

nonlinear susceptibilities | interference | Euler transformation | local field factor | molecular polarizability | Fresnel factor | AIR-WATER-INTERFACE | LIQUID INTERFACES | CHIRAL LIQUIDS | UNIFIED TREATMENT | CHEMISTRY, PHYSICAL | CATALYTIC-REACTIONS | CH STRETCHING MODES | MULTIPOLAR CONTRIBUTIONS | FORBIDDEN NATURE | OPTICAL 2ND-HARMONIC GENERATION | KLEINMAN SYMMETRY | Molecular dynamics | Models | Polarization (Light) | Quantum chemistry | Spectrum analysis | Methods

nonlinear susceptibilities | interference | Euler transformation | local field factor | molecular polarizability | Fresnel factor | AIR-WATER-INTERFACE | LIQUID INTERFACES | CHIRAL LIQUIDS | UNIFIED TREATMENT | CHEMISTRY, PHYSICAL | CATALYTIC-REACTIONS | CH STRETCHING MODES | MULTIPOLAR CONTRIBUTIONS | FORBIDDEN NATURE | OPTICAL 2ND-HARMONIC GENERATION | KLEINMAN SYMMETRY | Molecular dynamics | Models | Polarization (Light) | Quantum chemistry | Spectrum analysis | Methods

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2019, Volume 346, pp. 594 - 611

In this paper we present a new family of identities for Euler sums and integrals of polylogarithms by using the methods of generating function and integral...

Euler sum | Riemann zeta function | Multiple zeta (star) value | Multiple harmonic (star) sum | polylogarithm function | Harmonic number | INTEGRALS | MATHEMATICS, APPLIED | MULTIPLE ZETA VALUES

Euler sum | Riemann zeta function | Multiple zeta (star) value | Multiple harmonic (star) sum | polylogarithm function | Harmonic number | INTEGRALS | MATHEMATICS, APPLIED | MULTIPLE ZETA VALUES

Journal Article

BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY, ISSN 0126-6705, 01/2020, Volume 43, Issue 1, pp. 847 - 877

Let p, p(1),..., p(m) be positive integers with p(1) <= p(2) <= center dot center dot center dot <= p(m) and x is an element of [-1, 1), define the so-called...

MATHEMATICS | Polylogarithm function | IDENTITIES | Harmonic number | Multiple harmonic sum | Euler sum | Riemann zeta function | Multiple zeta value | EXPLICIT EVALUATION | MULTIPLE ZETA VALUES

MATHEMATICS | Polylogarithm function | IDENTITIES | Harmonic number | Multiple harmonic sum | Euler sum | Riemann zeta function | Multiple zeta value | EXPLICIT EVALUATION | MULTIPLE ZETA VALUES

Journal Article