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A special constant and series with zeta values and harmonic numbers
Abstract
In this paper we demonstrate the importance of a mathematical constant which is the value of several interesting numerical series involving harmonic numbers, zeta values, and logarithms. We also evaluate in closed form a number of numerical and power series.
... This estimate follows from (1) and a similar estimate for ζ(n + 1) − 1 shown in [5]. ...
... The constant K exhibits strong similarity to another constant, M , that was defined and studied by the first author [5]: ...
... In this section it will be proved that several interesting eta series can be evaluated in terms of the new constant K. The results from the next theorem are similar to those of [5,Theorem 2]. We recall that harmonic numbers (H n ) n≥0 are defined by ...
In this article, we evaluate in closed form a number of series involving values of the Dirichlet eta function, and also Fibonacci and Lucas numbers. We also introduce a special constant representing the values of several such series.
... Proposition A: Let f (z) be a function analytic in a region of the form Re(z) > λ for some λ < 0 and with moderate growth in that region. Then we have the representation n k (−1) k f (k) (5) and in particular, ...
... The case r = 1 was discussed in [5]. Setting m = 1 in (37) ...
... The first integral when r = 1 is (This complements the results in [5]). ...
We evaluate in closed form several series involving products of Cauchy numbers with other special numbers (harmonic, skew-harmonic, hyperharmonic, and central binomial). Similar results are obtained with series involving Stirling numbers of the first kind. We focus on several particular cases which give new closed forms for Euler sums of hyperharmonic numbers and products of hyperharmonic and harmonic numbers.
... This integral also occurs in [6]: Please see [6] for more represantations of this integral. ...
... This integral also occurs in [6]: Please see [6] for more represantations of this integral. ...
We evaluate in closed form several series involving products of Cauchy numbers with other special numbers (harmonic, skew-harmonic, hyperharmonic, and central binomial). Similar results are obtained with series involving Stirling numbers of the first kind. We focus on several particular cases which give new closed forms for Euler sums of hyperharmonic numbers and products of hyperharmonic and harmonic numbers.
... In [10] the author introduced the constant ...
... x a s , the Riemann zeta function () s , and the polylogarithm Li ( ) s x . We also proved a new representation of Euler's constant and a new representation of the constant M introduced by the autor in [10]. ACNOWLEDGEMENT The autor is thankful to the referee for a number of valuable suggestions that helped to improve the paper. ...
We prove a short general theorem which immediately implies some classical results of Hasse, Guillera and Sondow, Paolo Amore, and also Alzer and Richards. At the end we obtain a new representation for the Euler constant gamma. The theorem transforms every Taylor series into a series depending on a parameter.
... In [10] the author introduced the constant ...
... x a s , the Riemann zeta function () s , and the polylogarithm Li ( ) s x . We also proved a new representation of Euler's constant and a new representation of the constant M introduced by the autor in [10]. ACNOWLEDGEMENT The autor is thankful to the referee for a number of valuable suggestions that helped to improve the paper. ...
We prove a short general theorem which immediately implies some classical results of Hasse, Guillera and Sondow, Paolo Amore, and also Alzer and Richards. At the end we obtain a new representation for Euler’s constant γ. The theorem transforms every Taylor series into a series depending on a parameter.
... In order to do this, we consider a natural generalization of the Roman harmonic numbers that were first introduced in [5] (see Definition 2), and use a general transformation formula that links the Cauchy numbers to the Ramanujan summation of series [3,Theorem 18]. A noteworthy fact is the presence in most of our formulas of certain alternating series with zeta values (see Definition 3) that recently appeared in different contexts [1,6,7]. In the aim to help the reader to find his way among our various formulas, a summary of the most noteworthy identities, ranked in ascending order of complexity, is given in the penultimate section of the article. ...
... For r = 1, the harmonic numbers H (1) n,k will be noted H n,k in the remainder of the article. Remark 1. ...
In this article, we present a class of identities linking together Cauchy numbers, the special values of the Riemann zeta function and its derivative, and a generalization of the Roman harmonic numbers, which represents a significant refinement and improvement of our earlier work on the subject.
... has been thoroughly studied by Boyadzhiev [4] who provided several remarkable formulas (see also [8, p. 142] and [9, p. 1836]). The series τ p for p = 2, 3, 4, . . . ...
... The series τ p for p = 2, 3, 4, . . . have been extensively studied in [11], they also appear in [4] and [10]. ...
In this study, we determine certain constants which naturally occur in the Laurent expansion of harmonic zeta functions.
... The constant ν −1,0 also arises in several other contexts in number theory. It appears [10] in the asymptotic formula for the number of divisors of n!, several series [3,8] relate it to the Riemann zeta function and to harmonic numbers, and it occurs in certain Ramanujan summations [6,7] involving harmonic numbers. Among its many series expressions we find by means of (2.14). ...
We demonstrate that the multiple Hurwitz zeta function is the ordinary generating function for the sequence of height 1 multiple zeta functions. This principle is then used to evaluate various series involving such zeta functions and other important sequences of number theoretic and combinatorial nature.
... The constant τ 1 ≈ 1.257746 . . . also appears in several contexts in number theory [6,9,13,14], having several other interesting expressions including ...
We use everywhere-convergent series for the height 1 multiple zeta functions to determine the singular parts of their Laurent series at each of their poles, and give an expression for each first “Stieltjes constant” (i.e., the linear Laurent coefficient) as series involving the Bernoulli numbers of the second kind, generalizing the classical Mascheroni series for Euler’s constant . The first Stieltjes constants at s=1 and at s=0 are then interpreted in terms of the Ramanujan summation of multiple harmonic star sums .
... in terms of the zeta values, Euler-Mascheroni constant, Stieltjes constant and some other certain constants (Theorem 7 or more pricesely Eq. (18) with Remark 3), for instance [7,13,18,20]). ...
In this paper, we consider meromorphic extension of the function (which we call hyperharmonic zeta function) where are the hyperharmonic numbers. We establish certain constants, denoted , which naturally occur in the Laurent expansion of . Moreover, we show that the constants and integrals involving the generalized exponential integral can be written as a finite combination of some special constants.
... Series with terms similar to ζpnq´1 but different coefficients were studied by [BBC00,Boy18,Bri12,Can17,FV92,Sri88]. ...
We explain how several results of Ramanujan follow from the formalism of order series. Consider the operad generated by a binary associative and commutative operation and a binary associative operation, order series are one of the algebras over this operad. In our interpretation, Ramanujan worked with series inheriting the structure of the disjoint union of posets. We then provide new conceptual proofs of a couple of results by Ramanujan and describe a version of his results that depends on a choice of a series parallel poset.
... The following examples demonstrate Theorem 13: [6,12,17,19]). ...
In this paper, we consider meromorphic extension of the function (which we call \textit{hyperharmonic zeta function}) where are the hyperharmonic numbers. We establish certain constants, denoted , which naturally occur in the Laurent expansion of . Moreover, we show that the constants and integrals involving generalized exponential integral can be written as a finite combination of some special constants.
... The constant σ p occurs in several series and integral evaluations (see [7,12,17,21]). ...
In this paper, we present two new generalizations of the Euler-Mascheroni constant arising from the Dirichlet series associated to the hyperharmonic numbers. We also give some inequalities related to upper and lower estimates, and evaluation formulas.
... The constant 1 M is important, because it represents the value of some interesting series. It was shown in [7] are the skew harmonic numbers. Now we have one more series in this list. ...
In this article we use an interplay between Newton series and binomial formulas in order to generate a number of series identities involving Cauchy numbers, harmonic numbers, Laguerre polynomials, and Stirling numbers of the first kind.
Using the Ramanujan summation method, we derive some unusual formulas for a class of Euler sums (including divergent Euler sums) similar to the classical relations due to Euler.
The paper presents formulae for certain series involving the Riemann zeta function. These formulae are generalizations, in a natural way, of well known formulae, originating from Leonhard Euler. Formulaes that existed only for initial values are now found for every natural n. Relevant connections with various known results are also pointed out.
We propose and develop yet another approach to the problem of summation of series involving the Riemann Zeta function (s), the (Hurwitz's) generalized Zeta function (s; a), the Polygamma function (p) (z) (p = 0; 1; 2;), and the polylogarithmic function Li s (z). The key ingredients in our approach include certain known integral representations for (s) and (s; a). The method developed in this paper is illustrated by numerous examples of closed-form evaluations of series of the aforementioned types; the method developed in Section 2, in particular, has been implemented in Mathematica (Version 3.0). Many of the resulting summation formulas are believed to be new.
The integral representation of the Hadamard product of two functions is used to prove several Euler-type series transformation formulas. As applications we obtain three binomial identities involving harmonic numbers and an identity for the Laguerre polynomials. We also evaluate in a closed form certain power series with harmonic numbers
We demonstrate a visualization of Euler’s series for 1 − γ, where γ is the Euler-Mascheroni constant.
This is a unique book for studying special functions through zeta-functions. Many important formulas of special functions scattered throughout the literature are located in their proper positions and readers get enlightened access to them in this book. The areas covered include: Bernoulli polynomials, the gamma function (the beta and the digamma function), the zeta-functions (the Hurwitz, the Lerch, and the Epstein zeta-function), Bessel functions, an introduction to Fourier analysis, finite Fourier series, Dirichlet L-functions, the rudiments of complex functions and summation formulas. The Fourier series for the (first) periodic Bernoulli polynomial is effectively used, familiarizing the reader with the relationship between special functions and zeta-functions. © 2007 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.
Galambos has conjectured that the logarithm of the geometric mean of partial denominators in a Lüroth representation of a number between zero and one is a constant. We here prove that this constant is ≈0.7884 and plays the same role as Khintchine's constant for usual continued fraction expansions.
The object of the present paper is to investigate systematically several interesting families of summation formulas involving infinite series of the Riemann zeta function. Many of the various results, which are unified (and generalized) here in a remarkably simple manner, have received considerable attention in recent years. We also present a brief account of a number of analogous results associated with the (Hurwitz's) generalized zeta function.
v.1.Fundamental Algoriths -- v. 2. Seminumerican algorithms -- v. 3. Sorting and searching -- v. 4. Combinatorial algorithms -- v. 5. Synbtactical algorith -- v. 6. Theory of languages -- v. 7. Compilers
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