Evaluation of binomial series with harmonic numbers
We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.
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arXiv subjects
Khristo N. Boyadzhiev
Publications and source records attributed to Khristo N. Boyadzhiev.
Dirichlet series and series with Stirling numbers
This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers, derangement numbers, binomial coefficients, central binomial coefficients, and Catalan numbers.
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Several classical identities via Mellin's transform
We present a summation rule using the Mellin transform to give short proofs of some important classical relations between special functions and Bernoulli and Euler polynomials. For example, the values of the Hurwitz zeta function at the negative integers are expressed in terms of Bernoulli polynomials. We also show identities involving exponential and Hermite polynomials.
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Representations of functions in series with parameter
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.
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Bernoulli, poly-Bernoulli, and Cauchy polynomials in terms of Stirling and r-Stirling numbers
We review and discuss some results on the representation of Bernoulli, poly-Bernoulli numbers, and Bernoulli and Cauchy polynomials in terms of Stirling numbers of the first or second kind, or in terms of r-Stirling numbers.
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Stirling numbers and inverse factorial series
We study inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such numbers. Using various identities for Stirling numbers of the first kind we construct a number of expansions of functions in terms of inverse factorial series where the coefficients are special numbers. These results are used to prove/reprove the asymptotic expansion of some classical functions. We also prove a binomial formula involving inverse factorials.
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A binomial formula for evaluating integrals
In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging integrals.
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New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials
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.
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Identities for squared central binomial coefficients
We prove four identities for the squared central binomial coefficients. The first three of them reflect certain transformation properties of the complete elliptic integrals of the first and the second kind, while the last one is based on properties of the Lagrange polynomials.
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The spiral vortex. A story about tornados and bathtubs
In this paper, we show that equiangular spirals (also known as logarithmic spirals) appear naturally in fluids going through sinks. We provide a simple mathematical model that explains the spiral forms in some natural formations. For this model we need only small parts from the theories of complex variables and differential equations. The purpose of this educational paper is to stimulate students for further studies in fluid theory and differential equations.
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Exotic series with Bernoulli, harmonic, Catalan, and Stirling numbers
In this paper, we present a formula for generating various exotic series in the spirit of Ovidiu Furdui and Alina Sintamarian. Our new series (evaluated in closed form) involve Bernoulli, harmonic, and Catalan numbers. Also Stirling numbers of the second kind, other special numbers, and exponential polynomials. The results include series identities with Laguerre polynomials and derangement polynomials.
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Sums of powers and special polynomials
In this paper, we discuss sums of powers of the positive integers and compute both the exponential and ordinary generating functions for these sums. We express these generating functions in terms of exponential and geometric polynomials and also show their connection to other interesting series. In particular, we show their connection to an interesting problem of Ovidiu Furdui.
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Convolutions for Stirling numbers, Lah numbers, and binomial coefficients
We present various identities in the form of convolutions involving Stirling numbers of both kinds, Lah numbers, and binomial coefficients. Certain convolution polynomials are discussed also. The proofs are based on several series transformation formulas.
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New series with Cauchy and Stirling numbers, Part 2
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.
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Power series with inverse binomial coefficients and harmonic numbers
We construct the generating function for products of inverse central binomial coefficients with harmonic numbers.
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New identities with Stirling, hyperharmonic, and derangement numbers, Bernoulli and Euler polynomials, powers, and factorials
A number of identities are proved by using Stirling transforms. These identities involve Stirling numbers of the first and second kinds, hyperharmonic and derangement numbers, Bernoulli and Euler numbers and polynomials, powers, power sums, and factorials.
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Geometric Polynomials: Properties and Applications to Series with Zeta Values
We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to obtain a closed form evaluation of certain series involving Riemann's zeta function.
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A special constant and series with zeta values and harmonic numbers
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.
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