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Iaroslav V. Blagouchine

Publications and source records attributed to Iaroslav V. Blagouchine.

9 recordsLinked to original sources
arXiv2024

On a generalization of Watson's trigonometric sum (on Dowker's sum of order one half)

In this paper we study the finite trigonometric sum $\sum a_l\csc\big(πl/n\big)$, where $a_l$ are equal to $\cos(2πl ν/n)$ and where the summation index $l$ and the discrete parameter $ν$ both run through $1$ to $n-1$. This sum is a generalization of Watson's trigonometric sum, which has been extensively studied in a series of previous papers, and also may be regarded as the so-called Dowker sum of order one half. It occurs in various problems in mathematics, physics and engineering, and plays an important role in some number-theoretic problems. In the paper, we obtain several integral and series representations for the above-mentioned sum, investigate its properties, derive various, including asymptotic, expansions for it, and deduce very accurate upper and lower bounds for it (both bounds are asymptotically vanishing). In addition, we obtain two relatively simple approximate formulae containing only several terms, which are also very accurate and can be particularly appreciated in applications. Finally, we also derive several advanced summation formulae for the digamma functions, which relate the gamma and the digamma functions, the investigated sum, as well as the product of a sequence of cosecants $\prod\big(\csc(πl/n)\big)^{\csc(πl/n)}$

math.NT↗
arXiv2024

On a finite sum of cosecants appearing in various problems

In this paper we investigate the finite sum of cosecants $\sum\csc\big(φ+aπl/n\big),$ where the index $l$ runs through 1 to $n-1$ and $φ$ and $a$ are arbitrary parameters, as well as several closely related sums, such as similar sums of a series of secants, of tangents and of cotangents. These trigonometric sums appear in various problems in mathematics, physics, and a variety of related disciplines. Their particular cases were fragmentarily considered in previous works, and it was noted that even a simple particular case $\sum\csc\big(πl/n\big)$ does not have a closed-form, i.e. a compact summation formula. In the paper, we derive several alternative representations for the above-mentioned sums, study their properties, relate them to many other finite and infinite sums, obtain their complete asymptotic expansions for large $n$ and provide accurate upper and lower bounds (e.g. the typical relative error for the upper bound is lesser than $2\times10^{-9}$ for $n\geqslant10$ and lesser than $7\times10^{-14}$ for $n\geqslant50$, which is much better than the bounds we could find in previous works). Our researches reveal that these sums are deeply related to several special numbers and functions, especially to the digamma function (furthermore, as a by-product, we obtain several interesting summations formulae for the digamma function). Asymptotical studies show that these sums may have qualitatively different behaviour depending on the choice of $φ$ and $a$; in particular, as $n$ increases some of them may become sporadically large. Finally, we also provide several historical remarks related to various sums considered in the paper. We show that some results in the field either were rediscovered several times or can easily be deduced from various known formulae, including some formulae dating back to the XVIIIth century.

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arXiv2018

Three notes on Ser's and Hasse's representations for the zeta-functions

This paper is devoted to Ser's and Hasse's series representations for the zeta-functions, as well as to several closely related results. The notes concerning Ser's and Hasse's representations are given as theorems, while the related expansions are given either as separate theorems or as formulae inside the remarks and corollaries. In the first theorem, we show that the famous Hasse's series for the zeta-function, obtained in 1930 and named after the German mathematician Helmut Hasse, is equivalent to an earlier expression given by a little-known French mathematician Joseph Ser in 1926. In the second theorem, we derive a similar series representation for the zeta-function involving the Cauchy numbers of the second kind. In the third theorem, with the aid of some special polynomials, we generalize the previous results to the Hurwitz zeta-function. In the fourth theorem, we obtain a similar series with Gregory's coefficients of higher order. In the fifth theorem, we extend the results of the third theorem to a class of Dirichlet series. As a consequence, we obtain several globally convergent series for the zeta-functions. We also show that Hasse's series may be obtained much more easily by using the theory of finite differences, and we demonstrate that there exist numerous series of the same nature. In the sixth theorem, we show that Hasse's series is a simple particular case of a more general class of series involving the Stirling numbers of the first kind. All the expansions derived in the paper lead, in turn, to the series expansions for the Stieltjes constants, including new series with rational terms for Euler's constant, for the logarithm of the gamma-function, for the digamma and trigamma functions. Finally, in the Appendix, we prove an interesting integral representation for the Bernoulli polynomials of the second kind, formerly known as the Fontana-Bessel polynomials.

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arXiv2017

A note on some constants related to the zeta-function and their relationship with the Gregory coefficients

In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the so-called Gregory coefficients, which are also known as (reciprocal) logarithmic numbers, Cauchy numbers of the first kind and Bernoulli numbers of the second kind. In addition, two interesting series with rational terms are given for Euler's constant and the constant ln(2*pi), and yet another generalization of Euler's constant is proposed and various formulas for the calculation of these constants are obtained. Finally, in the paper, we mention that almost all the constants considered in this work admit simple representations via the Ramanujan summation.

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arXiv2017

A Note on Some Recent Results for the Bernoulli Numbers of the Second Kind

In a recent issue of the Bulletin of the Korean Mathematical Society, Qi and Zhang discovered an interesting integral representation for the Bernoulli numbers of the second kind (also known as Gregory's coefficients, Cauchy numbers of the first kind, and the reciprocal logarithmic numbers). The same representation also appears in many other sources, either with no references to its author, or with references to various modern researchers. In this short note, we show that this representation is a rediscovery of an old result obtained in the XIXth century by Ernst Schröder. We also demonstrate that the same integral representation may be readily derived by means of complex integration. Moreover, we discovered that the asymptotics of these numbers were also the subject of several rediscoveries, including very recent ones. In particular, the first-order asymptotics, which are usually (and erroneously) credited to Johan F. Steffensen, actually date back to the mid-XIXth century, and probably were known even earlier.

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arXiv2016

Expansions of generalized Euler's constants into the series of polynomials in $π^{-2}$ and into the formal enveloping series with rational coefficients only

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $γ_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $π^{-2}$ with rational coefficients and converges slightly better than Euler's series $\sum n^{-2}$. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's constants, which is more accurate than several well-known estimations. Finally, in Appendix A, the reader will also find two simple integral definitions for the Stirling numbers of the first kind, as well an upper bound for them.

math.NT↗
arXiv2016

Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to $π^{-1}$

In this paper, two new series for the logarithm of the $Γ$-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling numbers of the first kind and have the property to contain only rational coefficients for certain arguments related to $π^{-1}$. In particular, for any value of the form $\lnΓ(\frac{1}{2}n\pmαπ^{-1})$ and $Ψ_k(\frac{1}{2}n\pmαπ^{-1})$, where $Ψ_k$ stands for the $k$th polygamma function, $α$ is positive rational greater than $\frac{1}{6}π$, $n$ is integer and $k$ is non-negative integer, these series have rational terms only. In the specified zones of convergence, derived series converge uniformly at the same rate as $\sum(n\ln^m\!n)^{-2}$, where $m=1, 2, 3,\ldots$\,, depending on the order of the polygamma function. Explicit expansions into the series with rational coefficients are given for the most attracting values, such as $\lnΓ(π^{-1})$, $\lnΓ(2π^{-1})$, $\lnΓ(\tfrac{1}{2}+π^{-1})$, $Ψ(π^{-1})$, $Ψ(\tfrac{1}{2}+π^{-1})$ and $Ψ_k(π^{-1})$. Besides, in this article, the reader will also find a number of other series involving Stirling numbers, Gregory's coefficients (logarithmic numbers, also known as Bernoulli numbers of the second kind), Cauchy numbers and generalized Bernoulli numbers. Finally, several estimations and full asymptotics for Gregory's coefficients, for Cauchy numbers, for certain generalized Bernoulli numbers and for certain sums with the Stirling numbers are obtained. In particular, these include sharp bounds for Gregory's coefficients and for the Cauchy numbers of the second kind.

math.NT↗
arXiv2015

A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations

Recently, it was conjectured that the first generalized Stieltjes constant at rational argument may be always expressed by means of Euler's constant, the first Stieltjes constant, the $Γ$-function at rational argument(s) and some relatively simple, perhaps even elementary, function. This conjecture was based on the evaluation of $γ_1(1/2)$, $γ_1(1/3)$, $γ_1(2/3)$, $γ_1(1/4)$, $γ_1(3/4)$, $γ_1(1/6)$, $γ_1(5/6)$, which could be expressed in this way. This article completes this previous study and provides an elegant theorem which allows to evaluate the first generalized Stieltjes constant at any rational argument. Several related summation formulæ involving the first generalized Stieltjes constant and the Digamma function are also presented. In passing, an interesting integral representation for the logarithm of the $Γ$-function at rational argument is also obtained. Finally, it is shown that similar theorems may be derived for higher Stieltjes constants as well; in particular, for the second Stieltjes constant the theorem is provided in an explicit form.

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