Levent Kargin

• Lecturer
• Research Assistant at Akdeniz University

We give closed-form evaluation formulas for the real and imaginary parts of the series $\sum_{m,n=1}^{\infty}\frac{e^{2\pi i\left( mx-ny\right) }} {m^{p}n^{r}\left( mc+n\right) ^{q}},$ $c\in\mathbb{N},$ in terms of certain zeta values.\textbf{ }Particular choices of $x$ and $y$ lead evaluation formulas for some Tornheim type $\sum_{m,n=1}^{\infty}\...

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

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 hyperharmo...

In this paper, we consider meromorphic extension of the function $$\begin{aligned} \zeta _{h^{\left( r\right) }}\left( s\right) =\sum _{k=1}^{\infty } \frac{h_{k}^{\left( r\right) }}{k^{s}},\text { }{\text {Re}}\left( s\right) >r \end{aligned}$$(which we call hyperharmonic zeta function) where \(h_{n}^{(r)}\) are the hyperharmonic numbers. We estab...

We obtain new recurrence relations, an explicit formula, and convolution identities for higher-order geometric polynomials. These relations generalize known results for geometric polynomials and lead to congruences for higher-order geometric polynomials and, in particular, for p-Bernoulli numbers.

In this paper, we examine the semiorthogonality of geometric and higher order geometric polynomials. As applications, we exhibit new explicit formulas for Bernoulli and p-Bernoulli numbers.

This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers H(p,q)n ?H(p,q)(r) = ?Xn=1 H(p,q)n/nr 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 the Riem...

This paper presents the evaluation of the Euler sums of generalized hyper-harmonic numbers Hn(p,q) 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 the Riemann zeta values.

In this paper, we consider meromorphic extension of the function \[ \zeta_{h^{\left( r\right) }}\left( s\right) =\sum_{k=1}^{\infty} \frac{h_{k}^{\left( r\right) }}{k^{s}},\text{ }\operatorname{Re}\left( s\right) >r, \] (which we call \textit{hyperharmonic zeta function}) where $h_{n}^{(r)}$ are the hyperharmonic numbers. We establish certain const...

UDC 517.5We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for -Bernoulli numbers.

In this paper, we present two new generalizations of the Euler-Mascheroni constant arising from the Dirichlet series associated to the hyperharmonic numbers. We also give some inequalities related to upper and lower estimates, and evaluation formulas.

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-ty...

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-ty...

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 hyperharmo...

In this study we introduce a second type of higher order generalized geometric polynomials. This we achieve by examining the generalized stirling numbers S(n, k, α, β, γ) [Hsu and Shiue, 1998] for some negative arguments. We study their number theoretic properties, asymptotic properties, and their combinatorial properties using the notion of barred...

In this paper, polynomials whose coefficients involve $r$-Lah numbers are used to evaluate several summation formulae involving binomial coefficients, Stirling numbers, harmonic or hyperharmonic numbers. Moreover, skew-hyperharmonic number is introduced and its basic properties are investigated.

Harmonic numbers and some of their generalizations are related to Bernoulli numbers and polynomials. In this paper, we present a connection between generalized hyperharmonic numbers and poly-Bernoulli polynomials. This relationship yields numerous identities for hyper-sums, series involving zeta values, and several congruences.

This paper is concerned with both kinds of the Cauchy numbers and their generalizations. Taking into account Mellin derivative, we relate p-Cauchy numbers of the second kind with shifted Cauchy numbers of the first kind, which yields new explicit formulas for the Cauchy numbers of the both kind. We introduce a generalization of the Cauchy numbers a...

The generalized hyperharmonic numbers H_{n}^{(p,q)} are defined by means of both the generalized harmonic and the hyperharmonic numbers. We evaluate the Euler sums of generalized hyperharmonic numbers in terms of the Euler sums of generalized harmonic numbers. Furthermore, we give evaluation formulas for several other series involving generalized h...

This paper gives new explicit formulas for sums of powers of integers and their reciprocals.

In this study we introduce a second type of higher order generalised geometric polynomials. This we achieve by examining the generalised stirling numbers $S(n; k;\alpha;\beta;\gamma)$ [Hsu & Shiue,1998] for some negative arguments. We study their number theoretic properties, asymptotic properties, and study their combinatorial properties using the...

In this paper, we give explicit expressions for generalized Apostol-Bernoulli and Apostol-Euler polynomials. As consequences, we deduce some explicit representations for other Apostol-type polynomials. Moreover, we find an algorithm based on a three-term recurrence for the calculation of generalized Apostol-Euler numbers and polynomials. As an appl...

In this paper, we take advantage of the Mellin type derivative to produce some new families of polynomials whose coefficients involve r-Lah numbers. One of these polynomials leads to rediscover many of the identities of r-Lah numbers. We show that some of these polynomials and hyperharmonic numbers are closely related. Taking into account of these...

We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order geometric polynomials, particularly for p-Bernoulli numbers.

In this paper, we first obtain several properties of poly-p-Bernoulli polynomials. In particular, we achieve some new results for poly-Bernoulli polynomials. We next define a generalization of the Arakawa–Kaneko zeta function associated with poly-p-Bernoulli polynomials, investigate some its particular values, and give asymptotic and series expansi...

In this note, we give an alternative proof of the generating function of $p$-Bernoulli numbers. Our argument is based on the Euler's integral representation.

In this paper, using geometric polynomials, we obtain a generating function of p-Bernoulli numbers in terms of harmonic numbers. As consequences of this generating function, we derive closed formulas for the finite summation of Bernoulli and harmonic numbers involving Stirling numbers of the second kind. We also give a relationship between the p-Be...

We relate geometric polynomials and p-Bernoulli polynomials with an integral representation, then obtain several properties of p-Bernoulli polynomials. These results yield new identities for Bernoulli numbers. Moreover, we evaluate a Faulhaber-type summation in terms of p-Bernoulli polynomials. Finally, we introduce poly-p-Bernoulli polynomials and...

In this paper, we evaluate sums and integrals of products of two geometric polynomials and obtain new explicit formulas for geometric polynomials and numbers. As a consequence of these results, we give new explicit formulas for p-Bernoulli numbers, Apostol-Bernoulli functions, and integrals of products of two Apostol-Bernoulli functions.

According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and Euler polynomials. Furthermore, we find new formulas for the Carlitz's (Carlitz) and Howard's (Howard2) finite...

In this paper we evaluate sums and integrals of products of Fubini polynomials and have new explicit formulas for Fubini polynomials and numbers. As a consequence of these results new explicit formulas for p-Bernoulli numbers and Apostol-Bernoulli functions are given. Besides, integrals of products of Apostol-Bernoulli functions are derived.

According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and Euler polynomials. Furthermore, we find new formulas for the Carlitz's (Carlitz) and Howard's (Howard2) finite...

In this paper we define generalized exponential polynomials by means of the generalization of the Mellin derivative (Formula presented.). We give different proofs for some known results and obtain a new recurrence relation and a new operator formula for generalized exponential polynomials. We also define generalized geometric polynomials by means o...

Published in Hacettepe Journal of Mathematics and Statistics, Volume 44 (2) (2015), 341 – 350

In this study, obtaining the matrix analog of the Euler's reflection formula for the classical gamma function we expand the domain of the gamma matrix function and give a infinite product expansion of sinpxP. Furthermore we define Riemann zeta matrix function and evaluate some other matrix integrals. We prove a functional equation for Riemann zeta...

In this paper, modified Laguerre matrix polynomials which appear as finite series solutions of second-order matrix differential equation are introduced. Some formulas related to an explicit expression, a three-term matrix recurrence relation and a Rodrigues formula are obtained. Several families of bilinear and bilateral generating matrix functions...

Aktas¸ et. al. in [3] introduced the generalized Humbert matrix polynomials (G-HMP) PA n .m;x;y;c/. In this paper we focus on some properties of these matrix polynomials such as matrix recurrence relations, matrix differential equation and an integral representation. We introduce generalized forms of operational rules associated with operators corr...

In this study we give addition theorem, multiplication theorem and summation formula for Hermite matrix polynomials. We write Hermite matrix polynomials as hypergeometric matrix functions. We also obtain a new generating function for Hermite matrix polynomials and using this function, we prove some new results and relations.

In this study, we give multiplication formula for generalized Euler polynomials of order α and obtain some explicit recursive formulas. The multiple alternating sums with positive real parameters a and b are evaluated in terms of both generalized Euler and generalized Bernoulli polynomials of order α. Finally we obtained some interesting special ca...