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Mehmet Cenkci

Publications and source records attributed to Mehmet Cenkci.

8 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

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.

math.NT↗
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↗
arXiv2007

Twisted Dedekind Type Sums Associated with Barnes' Type Multiple Frobenius-Euler l-Functions

The aim of this paper is to construct new Dedekind type sums. We construct generating functions of Barnes' type multiple Frobenius-Euler numbers and polynomials. By applying Mellin transformation to these functions, we define Barnes' type multiple l-functions, which interpolate Frobenius-Euler numbers at negative integers. By using generalizations of the Frobenius-Euler functions, we define generalized Dedekind type sums and prove corresponding reciprocity law. We also give twisted versions of the Frobenius-Euler polynomials and new Dedekind type sums and corresponding reciprocity law. Furthermore, by using p-adic q-Volkenborn integral and twisted (h,q)-Bernoulli functions, we construct p-adic (h,q)-higher order Dedekind type sums. By using relation between Bernoulli and Frobenius-Euler functions, we also define analogues of Hardy-Berndt type sums. We give some new relations related to to these sums as well.

math.NT↗