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Mümün Can

Publications and source records attributed to Mümün Can.

At least 19 recordsLinked to original sources
arXiv2026

Analytic properties of the parametric harmonic zeta function, with applications to harmonic Stieltjes constants

This paper investigates the analytic structure of the parametric harmonic zeta function $ζ_{H}\left(s, a, b\right)$, a Dirichlet series associated with generalized harmonic numbers. We establish its meromorphic continuation to the complex plane and determine the residues at all of its poles. We then derive a Taylor expansion for $ζ_{H}\left(s, a, b+t\right)$, leading to harmonic analogues and extensions of classical identities due to Landau, Singh-Verma, and Srivastava. We further develop a systematic theory of the associated harmonic Stieltjes constants by deriving explicit formulas, including previously unknown cases, together with a limit representation for the first-order constants. Finally, we construct the harmonic analogue of the classical digamma function, establish its principal analytic properties and its connection with the harmonic Stieltjes constants, and obtain Raabe-type formulas for the parametric harmonic zeta function, the harmonic Stieltjes constants, and the harmonic digamma function.

math.CV↗
arXiv2025

Partial Deranged Bell Numbers and Their Combinatorial Properties

We introduce a novel generalization of deranged Bell numbers by defining the partial deranged Bell numbers $w_{n,r}$, which count the number of set partitions of $\left[ n\right] $ with exactly $r$ fixed blocks, while the remaining blocks are deranged. This construction provides a unified framework that connects partial derangements, Stirling numbers, and ordered Bell numbers. We investigate their combinatorial properties, including explicit formulas, generating functions, and recurrence relations. Moreover, we demonstrate that these numbers are expressible in terms of classical sequences such as deranged Bell numbers and ordered Bell numbers, and reveal their relationship to complementary Bell numbers, offering insights relevant to Wilf's conjecture. Notably, we derive the identity \[ \tildeϕ_{n}=\Tilde{w}_{n,0}-\Tilde{w}_{n,1}=\tilde{w}_{n-1,0}-2\tilde {w}_{n-1,2}, \] which illustrates their structural connection to complementary Bell numbers. We also introduce a polynomial expansion for these numbers and explore their connections with exponential polynomials, geometric polynomials, and Bernoulli numbers. These relationships facilitate the derivation of closed-form expressions for certain finite summations involving Stirling numbers of the second kind, Bernoulli numbers, and binomial coefficients, articulated through partial derangement numbers.

math.CO↗
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.

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↗
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.

math.NT↗
arXiv2020

Reciprocity formulas for Hall-Wilson-Zagier type Hardy-Berndt sums

In this paper, we introduce vast generalizations of the Hardy-Berndt sums. They involve higher-order Euler and/or Bernoulli functions, in which the variables are affected by certain linear shifts. By employing the Fourier series technique we derive linear relations for these sums. In particular, these relations yield reciprocity formulas for Carlitz, Rademacher, Mikolás and Apostol type generalizations of the Hardy-Berndt sums, and give rise to generalizations for some Goldberg's three-term relations. We also present an elementary proof for the Mikolás' linear relation and a reciprocity formula in terms of the generation function.

math.NT↗
arXiv2019

Polynomials with r-Lah coefficient and hyperharmonic numbers

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 connections, we reach several identities for harmonic and hyperharmonic numbers.

math.NT↗
arXiv2017

Transformation formulas of a character analogue of $\logθ_{2}(z)$

In this paper, transformation formulas for the function \[ A_{1}\left(z,s:χ\right)=\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}χ\left(n\right)χ\left(m\right)\left(-1\right)^{m}n^{s-1}e^{2πimnz/k} \] are obtained. Sums that appear in transformation formulas are generalizations of the Hardy--Berndt sums $s_{j}(d,c),$ $j=1,2,5$. As applications of these transformation formulas, reciprocity formulas for these sums are derived and several series relations are presented.

math.NT↗
arXiv2016

Character analogue of the Boole summation formula with applications

In this paper, we present the character analogue of the Boole summation formula. Using this formula, an integral representation is derived for the alternating Dirichlet $L-$function and its derivative is evaluated at $s=0$. Some applications of the character analogue of the Boole summation formula and the integral representation are given about the alternating Dirichlet $L-$function. Moreover, the reciprocity formulas for two new arithmetic sums, arising from utilizing the summation formulas, and for Hardy--Berndt sum $S_{p}\left( b,c:χ\right) $ are proved.

math.NT↗
arXiv2016

On the integral of products of higher-order Bernoulli and Euler polynomials

In this paper, we derive a formula on the integral of products of the higher-order Euler polynomials. By the same way, similar relations are obtained for $l$ higher-order Bernoulli polynomials and $r$ higher-order Euler polynomials. Moreover, we establish the connection between the results and the generalized Dedekind sums and Hardy--Berndt sums. Finally, the Laplace transform of Euler polynomials is given.

math.NT↗
arXiv2016

Periodic analogues of Dedekind sums and transformation formulas of Eisenstein series

In this paper, a transformation formula under modular substitutions is derived for a large class of generalized Eisenstein series. Appearing in the transformation formulae are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity theorems are proved for these Dedekind sums. Furthermore, as an application of the transformation formulae, relations between various infinite series and evaluations of several infinite series are deduced. Finally, we consider these sums for some special cases.

math.NT↗
arXiv2015

Character analogues of certain Hardy-Berndt sums

In this paper we consider transformation formulas for \[ B\left( z,s:χ\right) =\sum\limits_{m=1}^{\infty}\sum\limits_{n=0} ^{\infty}χ(m)χ(2n+1)\left( 2n+1\right) ^{s-1}e^{πim(2n+1)z/k}. \] We derive reciprocity theorems for the sums arising in these transformation formulas and investigate certain properties of them. With the help of the character analogues of the Euler--Maclaurin summation formula we establish integral representations for the Hardy-Berndt character sums $s_{3,p}\left( d,c:χ\right) $ and $s_{4,p}\left( d,c:χ\right) $.

math.NT↗
arXiv2015

On reciprocity formula of character Dedekind sums and the integral of products of Bernoulli polynomials

We give a simple proof for the reciprocity formulas of character Dedekind sums associated with two primitive characters, whose modulus need not to be same, by utilizing the character analogue of the Euler-MacLaurin summation formula. Moreover, we extend known results on the integral of products of Bernoulli polynomials by considering the integral \[ \int\limits_{0}^{x}B_{n_{1}}(b_{1}z+y_{1})... B_{n_{r}}(b_{r}z+y_{r}) dz, \] where $b_{l}$ $(b_{l}\neq 0)$ and $y_{l}$ $(1\leq l\leq r)$ are real numbers. As a consequence of this integral we establish a connection between the reciprocity relations of sums of products of Bernoulli polynomials and of the Dedekind sums.

math.NT↗