arXiv subjects
Publications and source records attributed to Khristo N. Boyadzhiev.
Supportive attitudes can bring to a blossoming science, while neglect can quickly make science absent from everyday life and provide a very primitive view of the world. We compare one important Greek achievement, the computation of the Earth meridian by Eratosthenes, to its later interpretation by the Roman historian of science Pliny.
The derivative polynomials for the hyperbolic and trigonometric tangent, cotangent and secant are found in explicit form, where the coefficients are given in terms of Stirling numbers of the second kind. As application, some integrals are evaluated and the reflection formula for the polygamma function is written in explicit form.
This article is a survey of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned.
It is shown that the curious identity of Simons follows immediately from Euler's series transformation formula and also from an identity due to Ljunggren. The relation of Simons' identity to Legendre's polynomials is also discussed. At the end we use the generalized Euler series transformation to obtain two recent binomial identities of Munarini.
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
Five series are evaluated in terms of zeta values. Three of the series involve harmonic numbers and one involves Stirling numbers of the first kind. The evaluation of these series is reduced to the evaluation of certain integrals, including the moments of the polylogarithm.
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.
This is a short survey of a class of functions introduces by Tom Apostol. The survey is focused on their relation to Eulerian polynomials, derivative polynomials, and also on some integral representations.
We study a special Dirichlet series studied before by Apostol and Matsuoka and specify its values at negative integers. These values are related to a certain convolution of Bernoulli numbers
We discuss a special function (polyexponential) that extends the natural exponential function and also the exponential integral. The basic properties of the polyexponential are listed and some applications are given. In particular, it is shown that certain Mellin integrals can be evaluated in terms of polyexponentials. The polyexponential is related to the exponential polynomials, the Riemann zeta function, the Dirichlet eta function and the Lerch Transcendent.
We introduce a new technique for evaluation of series with zeta coefficients and also for evaluation of certain integrals involving the logGamma function. This technique is based on Hankel integral representations of the Hurwitz zeta, the Lerch Zeta, the Digamma and logGamma functions.