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Ayhan Dil

Publications and source records attributed to Ayhan Dil.

17 recordsLinked to original sources
arXiv2024

On certain harmonic zeta functions

This study deals with certain harmonic zeta functions, one of them occurs in the study of the multiplication property of the harmonic Hurwitz zeta function. The values at the negative even integers are found and Laurent expansions at poles are described. Closed-form expressions are derived for the Stieltjes constants that occur in Laurent expansions in a neighborhood of s=1. Moreover, as a bonus, it is obtained that the values at the positive odd integers of three harmonic zeta functions can be expressed in closed-form evaluations in terms of zeta values and log-sine integrals.

math.NT↗
arXiv2023

On the Stieltjes constants with respect to harmonic zeta functions

The aim of this paper is to investigate harmonic Stieltjes constants occurring in the Laurent expansions of the function \[ ζ_{H}\left( s,a\right) =\sum_{n=0}^{\infty}\frac{1}{\left( n+a\right) ^{s}}\sum_{k=0}^{n}\frac{1}{k+a},\text{ }\operatorname{Re}\left( s\right) >1, \] which we call harmonic Hurwitz zeta function. In particular evaluation formulas for the harmonic Stieltjes constants $γ_{H}\left( m,1/2\right) $ and $γ_{H}\left( m,1\right) $ are presented.

math.NT↗
arXiv2021

Stieltjes constants appearing in the Laurent expansion of the hyperharmonic zeta function

In this paper, we consider meromorphic extension of the function \[ ζ_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r, \] (which we call \textit{hyperharmonic zeta function}) where $h_{n}^{(r)}$ are the hyperharmonic numbers. We establish certain constants, denoted $γ_{h^{\left( r\right) }}\left( m\right) $, which naturally occur in the Laurent expansion of $ζ_{h^{\left( r\right) }}\left( s\right) $. Moreover, we show that the constants $γ_{h^{\left( r\right) }}\left( m\right) $ and integrals involving generalized exponential integral can be written as a finite combination of some special constants.

math.NT↗
arXiv2021

Generalized harmonic numbers via poly-Bernoulli polynomials

We present a relationship between the generalized hyperharmonic numbers and the poly-Bernoulli polynomials, motivated from the connections between harmonic and Bernoulli numbers. This relationship yields numerous identities for the hyper-sums and several congruences.

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arXiv2021

On Evaluations of Euler-type Sums of Hyperharmonic Numbers

We give explicit evaluations of the linear and non-linear Euler sums of hyperharmonic numbers $h_{n}^{\left( r\right) }$ with reciprocal binomial coefficients. These evaluations enable us to extend closed form formula of Euler sums of hyperharmonic numbers to an arbitrary integer $r$. Moreover, we reach at explicit formulas for the shifted Euler-type sums of harmonic and hyperharmonic numbers. All the evaluations are provided in terms of the Riemann zeta values, harmonic numbers and linear Euler sums.

math.NT↗
arXiv2021

Euler sums of generalized harmonic numbers and connected extensions

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers $H_{n}^{\left( p,q\right) }$ \[ ζ_{H^{\left( p,q\right) }}\left( r\right) =\sum\limits_{n=1}^{\infty }\dfrac{H_{n}^{\left( p,q\right) }}{n^{r}}% \] in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of Riemann zeta values.

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arXiv2020

Tornheim-like series, harmonic numbers and zeta values

Explicit evaluations of the Tornheim-like double series in the form \[ \sum_{n,m=1}^\infty \frac{H_{n+m+s}}{nm\left( n+m+s \right)},\ s\in \mathbb{N\cup } \left\{ 0 \right\} \] and their extensions are given. Furthermore, series of the type \[ \sum_{m=1}^\infty \frac{2H_{2m+1}-H_{m}}{2m\left( 2m+1 \right)} \] and some other Tornheim-like multiple series are evaluated in terms of the zeta values.

math.NT↗
arXiv2019

Applications of derivative and difference operators on some sequences

In this study, depending on the upper and the lower indices of the hyperharmonic number $h_{n}^{(r)}$, nonlinear recurrence relations are obtained. It is shown that generalized harmonic number and hyperharmonic number can be obtained from derivatives of the binomial coefficients. Taking into account of difference and derivative operators, several identities of the harmonic and hyperharmonic numbers are given. Negative-ordered hyperharmonic number is defined and its alternative representations are given.

math.NT↗
arXiv2017

Series with Hermite Polynomials and Applications

We obtain a series transformation formula involving the classical Hermite polynomials. We then provide a number of applications using appropriate binomial transformations. Several of the new series involve Hermite polynomials and harmonic numbers, Lucas sequences, exponential and geometric numbers. We also obtain a series involving both Hermite and Laguerre polynomials, and a series with Hermite polynomials and Stirling numbers of the second kind.

math.NT↗
arXiv2013

Euler Sums of Hyperharmonic Numbers

The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: σ(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mező and Dil. We also provide an explicit evaluation of σ(r,m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evaluate several other series involving hyperharmonic numbers.

math.NT↗
arXiv2008

A Symmetric Algorithm for Hyperharmonic and Fibonacci Numbers

In this work, we introduce a symmetric algorithm obtained by the recurrence relation a_{n}^{k}=a_{n-1}^{k}+a_{n}^{k-1}. We point out that this algorithm can be apply to hyperharmonic-, ordinary and incomplete Fibonacci- and Lucas numbers. An explicit formulae for hyperharmonic numbers, general generating functions of the Fibonacci- and Lucas numbers are obtained. Besides we define "hyperfibonacci numbers", "hyperlucas numbers". Using these new concepts, some relations between ordinary and incomplete Fibonacci- and Lucas numbers are investigated.

math.NT↗