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

Book

2018, 1st ed. 2018, ISBN 9783319746470, 168

This book develops the foundations of "summability calculus", which is a comprehensive theory of fractional finite sums...

Mathematical analysis | Finite differences | Fractional calculus | Ordinary Differential Equations | Special Functions | Sequences, Series, Summability | Approximations and Expansions | Mathematics | Number Theory | Real Functions | Series acceleration | Asymptotic expansions | Numerical integration | Bernoulli numbers | Euler-mascheroni constant | Divergent series | Gamma and polygamma functions | Euler-maclaurin summation formula | Analytic summability theory | Matrix summability methods | Riemann zeta function | Fractional finite sums | Special functions

Mathematical analysis | Finite differences | Fractional calculus | Ordinary Differential Equations | Special Functions | Sequences, Series, Summability | Approximations and Expansions | Mathematics | Number Theory | Real Functions | Series acceleration | Asymptotic expansions | Numerical integration | Bernoulli numbers | Euler-mascheroni constant | Divergent series | Gamma and polygamma functions | Euler-maclaurin summation formula | Analytic summability theory | Matrix summability methods | Riemann zeta function | Fractional finite sums | Special functions

eBook

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...

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

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 representations of series...

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

...) and x is an element of [-1, 1), define the so-called Euler-type sums Sp(1) p(2)... p(m), p (x), which are the infinite sums whose general term is a product of harmonic numbers of index n, a power of n...

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

MATHEMATICS, ISSN 2227-7390, 09/2019, Volume 7, Issue 9, p. 833

In this paper, we present some Euler-like sums involving partial sums of the harmonic and odd harmonic series...

SUMMATION FORMULAS | MATHEMATICS | SERIES | closed form | ArcTan and ArcTanh functions | Catalan's constant | HARMONIC SUMS | Euler sums | Trigamma function | integral representation | partial fractions | FAMILY | Catalan’s constant

SUMMATION FORMULAS | MATHEMATICS | SERIES | closed form | ArcTan and ArcTanh functions | Catalan's constant | HARMONIC SUMS | Euler sums | Trigamma function | integral representation | partial fractions | FAMILY | Catalan’s constant

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 11/2018, Volume 14, Issue 10, pp. 2699 - 2728

... of the gcd -sum function f ( gcd ( j , k ) ) and the function ∑ d | k , d s | j ( f ∗ μ ) ( d ) for any positive integers j and k , namely...

gcd -sum functions | mean value formula | Euler totient function | Dedekind function | MATHEMATICS | GENERALIZED RAMANUJAN SUMS | THEOREM | EXTENSION | gcd-sum functions

gcd -sum functions | mean value formula | Euler totient function | Dedekind function | MATHEMATICS | GENERALIZED RAMANUJAN SUMS | THEOREM | EXTENSION | gcd-sum functions

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 04/2017, Volume 13, Issue 3, pp. 655 - 672

In this paper, we work out some explicit formulae for double nonlinear Euler sums involving harmonic numbers and alternating harmonic numbers...

Euler sum | Riemann zeta function | Polylogarithm function | harmonic number | INTEGRALS | MATHEMATICS | VALUES

Euler sum | Riemann zeta function | Polylogarithm function | harmonic number | INTEGRALS | MATHEMATICS | VALUES

Journal Article

Journal of Difference Equations and Applications, ISSN 1023-6198, 07/2019, Volume 25, Issue 7, pp. 1007 - 1023

.... Using the decompositions, we discuss the evaluations of some Euler-type sums involving harmonic numbers and binomial coefficients, such as and...

Riemann zeta function | Euler-type sums | harmonic numbers | binomial coefficients | MATHEMATICS, APPLIED | SERIES | IDENTITIES | DUALITY | EULER SUMS | Binomial coefficients | Decomposition | Sums

Riemann zeta function | Euler-type sums | harmonic numbers | binomial coefficients | MATHEMATICS, APPLIED | SERIES | IDENTITIES | DUALITY | EULER SUMS | Binomial coefficients | Decomposition | Sums

Journal Article

Journal of the Korean Mathematical Society, ISSN 0304-9914, 2018, Volume 55, Issue 5, pp. 1207 - 1220

.... Moreover, we prove that the Euler-type sums with hyperharmonic numbers: S(k, m; p) := Sigma(infinity)(n=1) h(n)((m)) (k)/n(p) (p >= m + 1, k = 1, 2, 3) can be expressed...

Euler sums | Riemann zeta function | Generalized hyperharmonic numbers | Stirling numbers | Harmonic numbers | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | ZETA-FUNCTION | SERIES | generalized hyperharmonic numbers | VALUES | harmonic numbers

Euler sums | Riemann zeta function | Generalized hyperharmonic numbers | Stirling numbers | Harmonic numbers | INTEGRALS | MATHEMATICS | MATHEMATICS, APPLIED | ZETA-FUNCTION | SERIES | generalized hyperharmonic numbers | VALUES | harmonic numbers

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 05/2014, Volume 10, Issue 3, pp. 737 - 762

...}.$$ We derive reciprocity theorems for the sums arising in these transformation formulas and investigate certain properties...

Bernoulli polynomials | Hardy-Berndt sums | Euler-Maclaurin formula | Dedekind sums | MATHEMATICS | THETA-FUNCTIONS | ANALYTIC EISENSTEIN SERIES | Heterocyclic compounds | Mathematics - Number Theory

Bernoulli polynomials | Hardy-Berndt sums | Euler-Maclaurin formula | Dedekind sums | MATHEMATICS | THETA-FUNCTIONS | ANALYTIC EISENSTEIN SERIES | Heterocyclic compounds | Mathematics - Number Theory

Journal Article

Discrete Mathematics, ISSN 0012-365X, 2009, Volume 309, Issue 10, pp. 3346 - 3363

.... Particularly, some of these identities are also related to the power sums and alternate power sums...

Alternate power sums | Power sums | Bernoulli polynomials | Genocchi polynomials | Combinatorial identities | Euler polynomials | MATHEMATICS | SYMMETRY | SERIES | NUMBERS | RECURRENCE

Alternate power sums | Power sums | Bernoulli polynomials | Genocchi polynomials | Combinatorial identities | Euler polynomials | MATHEMATICS | SYMMETRY | SERIES | NUMBERS | RECURRENCE

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 08/2017, Volume 13, Issue 7, pp. 1695 - 1709

... ! , and then we produce the generating function and an integral representation for S n ( m ) . Using them we evaluate many interesting finite and infinite harmonic sums in closed form...

Harmonic sums | Bell polynomials | Boole's formula | generalized harmonic numbers | Riemann zeta function | Apery constant | harmonic numbers | Stirling numbers | combinatorial identities | MATHEMATICS | HYPERGEOMETRIC-SERIES | EULER | Mathematics - Number Theory

Harmonic sums | Bell polynomials | Boole's formula | generalized harmonic numbers | Riemann zeta function | Apery constant | harmonic numbers | Stirling numbers | combinatorial identities | MATHEMATICS | HYPERGEOMETRIC-SERIES | EULER | Mathematics - Number Theory

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 5/2015, Volume 37, Issue 1, pp. 89 - 108

In this paper, we first develop a set of identities for Euler-type sums. We then investigate products of the shifted harmonic numbers and the reciprocal binomial coefficients...

Psi (or Digamma) function | 33B15 | Primary 05A10 | Binomial coefficients | Functions of a Complex Variable | Secondary 11B83 | 33C20 | Field Theory and Polynomials | Mathematics | Fourier Analysis | Integral representations | Harmonic numbers | Euler sums | Polygamma functions | 05A19 | 11M06 | Number Theory | Riemann Zeta function | Combinatorics | SUMMATION FORMULAS | MATHEMATICS | SERIES | NUMBER SUMS

Psi (or Digamma) function | 33B15 | Primary 05A10 | Binomial coefficients | Functions of a Complex Variable | Secondary 11B83 | 33C20 | Field Theory and Polynomials | Mathematics | Fourier Analysis | Integral representations | Harmonic numbers | Euler sums | Polygamma functions | 05A19 | 11M06 | Number Theory | Riemann Zeta function | Combinatorics | SUMMATION FORMULAS | MATHEMATICS | SERIES | NUMBER SUMS

Journal Article

Journal of number theory, ISSN 0022-314X, 2018, Volume 185, pp. 160 - 193

In this paper, using the Bell polynomials and the methods of generating function and integration, we establish various mixed Euler sums and Stirling sums, and present a unified approach to determining...

Euler sums | Riemann zeta function | Bell polynomials | Stirling sums | Generating functions | SERIES | EXPLICIT EVALUATION | INTEGRALS | POLYLOGARITHMS | MATHEMATICS | RIEMANN ZETA-FUNCTION | VALUES | GENERALIZED HARMONIC NUMBERS

Euler sums | Riemann zeta function | Bell polynomials | Stirling sums | Generating functions | SERIES | EXPLICIT EVALUATION | INTEGRALS | POLYLOGARITHMS | MATHEMATICS | RIEMANN ZETA-FUNCTION | VALUES | GENERALIZED HARMONIC NUMBERS

Journal Article

Monatshefte für Mathematik, ISSN 1436-5081, 2016, Volume 182, Issue 4, pp. 957 - 975

In this paper, we discuss the analytic representations of q-Euler sums which involve q-harmonic numbers through q-polylogarithms, either linearly or nonlinearly, and give explicit formulae for several...

65B10 | 33B15 | 05A30 | 11M99 | 11M32 | q-Riemann zeta function | Mathematics, general | Mathematics | 11M06 | q-Polylogarithm function | q-Euler sums | 33D05 | MATHEMATICS | ZETA-FUNCTIONS | INTEGRAL-REPRESENTATIONS

65B10 | 33B15 | 05A30 | 11M99 | 11M32 | q-Riemann zeta function | Mathematics, general | Mathematics | 11M06 | q-Polylogarithm function | q-Euler sums | 33D05 | MATHEMATICS | ZETA-FUNCTIONS | INTEGRAL-REPRESENTATIONS

Journal Article

BULLETIN OF THE KOREAN MATHEMATICAL SOCIETY, ISSN 1015-8634, 07/2017, Volume 54, Issue 4, pp. 1255 - 1280

In this paper we present a new family of identities for parametric Euler sums which generalize a result of David Borwein et al. [2...

INTEGRALS | MATHEMATICS | SERIES | Euler sum | Riemann zeta function | harmonic number | Hurwitz zeta function

INTEGRALS | MATHEMATICS | SERIES | Euler sum | Riemann zeta function | harmonic number | Hurwitz zeta function

Journal Article

Analysis Mathematica, ISSN 0133-3852, 12/2017, Volume 43, Issue 4, pp. 687 - 707

In this paper, we use Abel’s summation formula to evaluate several quadratic and cubic sums of the form $${F_N}(A,B;x): = \sum\limits_{n = 1}^N {(A - {A_n})(B...

sequence | 11M32 | Analysis | tail | multiple zeta star value (mzsv) | Riemann zeta function | Mathematics | 11M06 | harmonic number | Abel’s summation formula | multiple zeta value (mzv) | MATHEMATICS | MULTIPLE HARMONIC SERIES | Abel's summation formula | VALUES | NONLINEAR EULER SUMS

sequence | 11M32 | Analysis | tail | multiple zeta star value (mzsv) | Riemann zeta function | Mathematics | 11M06 | harmonic number | Abel’s summation formula | multiple zeta value (mzv) | MATHEMATICS | MULTIPLE HARMONIC SERIES | Abel's summation formula | VALUES | NONLINEAR EULER SUMS

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 04/2012, Volume 218, Issue 16, pp. 8064 - 8076

Let a1,a2,… be a numerical sequence. As the main task of the paper, we consider the classical problem of computing the sum ∑n=1...

Summation of divergent series | Infinity | Conditionally convergent series | Euler product formula | MATHEMATICS, APPLIED | Universe | Mathematical models | Tasks | Computation | Sums

Summation of divergent series | Infinity | Conditionally convergent series | Euler product formula | MATHEMATICS, APPLIED | Universe | Mathematical models | Tasks | Computation | Sums

Journal Article

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