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## Publications

Publications (78)

In this paper, we give a definition of the p-adic Arakawa-Kaneko-Hamahata zeta functions. These zeta functions interpolate Hamahata's poly-Euler polynomials at non-positive integers. We prove the derivative formula, the difference equation and the reflection formula of these zeta functions. Furthermore, we also prove a sums of products identity and...

In this article we first introduce some results about the binomial, trinomial, and quadrinomial convolution sums of certain divisor functions and then we provide an identity for the multinomial convolution sums of the divisor function .

Let n be a positive integer. We investigate the sequences ((Sm(n))m, which concerns the iteration of the odd divisor function S, given by
In this article, we introduced and studied the following invariants: order, m-gonal shape number, type, convexity, and area of n derived from Sm(n). For m = 3, 4, 5, we classify m-gonal shape odd prime integers....

Background:
Propagation of photon signals in biological systems, such as neurons, accompanies the production of biophotons. The role of biophotons in a cell merits special attention due to its applicability in various optical systems.
Objective:
This work aims to investigate the time behavior of biophoton signals emitted from living systems in d...

In this study, we propose an effective method of easy and intuitive modeling of various types of multiple leaves from plants, including flowering plants and trees, and of naturally visualizing them. This method consists of two processes. The first is the procedural modeling of leaf venation patterns. The proposed method enables modeling of the grow...

It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. In this article, we consider relationship between fifth-order combinatoric convolution sums of divisor functions and Bernoulli polynomials. As applications of these identities, we give a concrete interpretation in terms of the pro...

The Hurwitz-type Euler zeta function is defined as a deformation of the
Hurwitz zeta function: \begin{equation*}
\zeta_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In
algebraic number theory, it represents a partial zeta function of cyclotomic
fields in one version of Stark's conjectures (see [12, p.4249, (6.13)]).
Its special ca...

The Apostol-Dedekind sum with quasi-periodic Euler functions is an analogue
of Apostol's definition of the generalized Dedekind sum with periodic Bernoulli
functions. In this paper, using the Boole summation formula, we shall obtain
the reciprocity formula for this sum.

In this paper, we give relationship between Bernoulli-Euler polynomials and convolution sums of divisor functions. First, we establish two explicit formulas for certain combinatoric convolution sums of divisor functions derived from Bernoulli and Euler polynomials. Second, as applications, we show five identities concerning the third and fourth-ord...

It is known that certain convolution sums using Liouville identity can be expressed as a combination of divisor functions and Bernoulli numbers. In this article we find seven combinatorial convolution sums derived from divisor functions and Bernoulli numbers.

In this paper, we study combinatoric convolution sums of divisor functions and get values of this sum when n=2^mp. We find that the value of this convolution sum is represented by a sum of powers of 2 and Bernoulli or Euler number.

We study combinatoric convolution sums of certain divisor functions involving even indices. We express them as a linear combination of divisor functions and Euler polynomials and obtain identities D 2 k (n) = (1 / 4) σ 2 k + 1,0 (n; 2) - 2 · 4 2 k σ 2 k + 1 (n / 4) - (1 / 2) [ d n, d ≡ 1 (4) { E 2 k (d) + E 2 k (d - 1) } + 2 2 k d n, d ≡ 1 (2) E 2...

We investigate the explicit evaluation for the sum ∑ (a,b,x,y)∈ℕ 4 ,ax+by=n,C(x,y) ab in terms of various divisor functions (where C(x,y) is the set of residue conditions on x and y) for various fixed C(x, y). We also obtain some identities and congruences as interesting applications.

It is known that certain combinatorial convolution sums involving two divisor functions product formulae of arbitrary level can be explicitly expressed as a linear combination of divisor functions. In this article we deal with cases for certain combinatorial convolution sums involving three, four, six and twelve divisor functions product formula an...

In this paper, we study the convolution sums involving odd divisor functions, and their relations to Weierstrass -functions.

We study the combinatoric convolution sums involving odd divisor functions, their relations to Bernoulli numbers, and some interesting applications.

Mahmudov (2012, 2013) introduced and investigated some q -extensions of the q -Bernoulli polynomials ℬ n , q α x , y of order α , the q -Euler polynomials ℰ n , q α x , y of order α , and the q -Genocchi polynomials 𝒢 n , q α x , y of order α . In this paper, we give some identities for ℬ n , q α x , y , 𝒢 n , q α x , y , and ℰ n , q α x , y and th...

Hahn (Rocky Mt. J. Math. 37:1593-1622, 2007) established three differential equations according to
,
, and
, which allows us to obtain the values of the formulas for
etc. Finally, by using the above equations, we derive the algebraic curves.
MSC:
11A67.

It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. One of the main goals in this paper is to establish combinatoric convolution sums for the divisor sums σˆs(n)=∑d|n(−1)nd−1ds. Finally, we find a formula of certain combinatoric convolution sums and Bernoulli polynomials.
MSC:
11A...

Motivated by some earlier Diophantine works on triangular numbers by Ljunggren and Cassels, we consider similar problems for general polygonal numbers.

It is known that certain combinatorial convolution sums involving divisor functions of “levels” 1 and 2 can be explicitly expressed as a linear combination of divisor functions. In this article we deal with a case for arbitrary level and obtain an explicit expression.

For $n\ge 3$ let $f(n)$ be the least positive integer $k$ such that $\binom
nk>\frac{2^n}{n+1}$. In this paper we investigate the properties of $f(n)$.

Let denote the sum of the s-th power of the positive divisors of N and with , > 0 and . In a celebrated paper [33], Ramanuja proved using elementary arguments. The coefficients' relation in this identity () motivated us to write this article. In this article, we found the convolution sums for odd and even divisor functions with , , and . If N is an...

In this paper, we study a distinction the two generating functions : and ( = 2, 4, 6, 8, 10, 12, 16), where is the number of representations of as the sum of squares. We also obtain some congruences of representation numbers and divisor function.

In this paper we derive some identities of symmetry related to higher order q-Euler polynomials by using the multivariate fermionic p-adic q-integral on Z(p). Furthermore, some of these identities are also related to the q-analogue of the alternating power sums and the multiplication theorem.

In this paper, we investigate the convolution sums
∑ ( a + b + c ) x = n a , ∑ a x + b y = n a b , ∑ a x + b y + c z = n a b c , ∑ a x + b y + c z + d u = n a b c d ,
where a , b , c , d , x , y , z , u , n ∈ N . Many new equalities and inequalities involving convolution sums, Bernoulli numbers and divisor functions have also been given.
MSC: 11A05...

In this note, we show that is a weighted polynomial ring if and only if N = 1, 2, 4, where is the graded ring of integral-weighted modular forms for the congruence subgroup .

In this paper, we consider several convolution sums, namely, (), (), and (), and establish certain identities involving their finite products. Then we extend these types of product convolution identities to products involving Faulhaber sums. As an application, an identity involving the Weierstrass -function, its derivative and certain linear combin...

We study convolution sums of certain restricted divisor functions in detail and present explicit evaluations in terms of usual divisor functions for some specific situations.

In order to model a variety of natural trees that are appropriate to outdoor terrains consisting of multiple trees, this study proposes a modeling method of new growth rules(based on the convolution sums of divisor functions). Basically, this method uses an existing growth-volume based algorithm for efficient management of the branches and leaves t...

This study proposes a novel procedural modeling method using convolution sums of divisor functions to model a variety of natural trees in a virtual ecosystem efficiently. The basic structure of the modeling method defines the growth grammar, including the branch propagation, a growth pattern of branches and leaves, and a process of growth deformati...

We are motivated by Ramanujan’s recursion formula for sums of the product of two Eisenstein series (Berndt in Ramanujan’s Notebook, Part II, 1989, Entry 14, p.332) and its proof, and also by Besge-Liouville’s convolution identity for the ordinary divisor function
(Williams in Number Theory in the Spirit of Liouville, vol. 76, 2011, Theorem 12.3)....

Utilizing the theory of elliptic curves over ℂ to the normalized lattice
, its connection to the Weierstrass ℘-functions and to the Eisenstein series
and
, we establish some arithmetic identities involving certain arithmetic functions and convolution sums of restricted divisor functions. We also prove some congruence relations involving certa...

Using p -adic integral, many new convolution identities involving Bernoulli, Euler and Genocchi numbers are given.
MSC: 11B68, 11S80.

Let sigma(s)(N) denote the sum of the sth powers of the positive divisors of a positive integer N and let sigma(s)(N) = Sigma(d vertical bar N)(-1)(d-1)d(s) with d, N, and s positive integers. Hahn [12] proved that 16 Sigma(k < N) (sigma) over tilde (k) (sigma) over tilde (3) (N - K) = -(sigma) over tilde (5) + 2(N -1) (sigma) over tilde (3) (N) +...

In this paper, we study the convolution sums involving restricted divisor functions, their generalizations, their relations to Bernoulli numbers, and some interesting applications.
MSC: 11B68, 11A25, 11A67, 11Y70, 33E99.

In this paper, we find the coefficients for the Weierstrass and (, , )-functions in terms of the arithmetic identities appearing in divisor functions which are proved by Ramanujan ([23]). Finally, we reprove congruences for the functions and in Hahn's article [11, Theorems 6.1 and 6.2].

Kim et al. (2012) introduced an interesting p-adic analogue of the
Eulerian polynomials. They studied some identities on the Eulerian polynomials in
connection with the Genocchi, Euler, and tangent numbers. In this paper, by
applying the symmetry of the fermionic p-adic q-integral on ℤ𝑝, defined by
Kim (2008), we show a
symmetric relation between...

In this article, we shall give a generalization of the formula
Σk =
1N-1σ1(2nk)σ3(2n(N-k)).

In 1958, L.J. Mordell provided the formula for the integral of the product of
two Bernoulli polynomials, he also remarked: "The integrals containing the
product of more than two Bernoulli polynomials do not appear to lead to simple
results." In this paper, we provide explicit formulas for the integral of the
product of $r$ Bernoulli polynomials, wh...

Reversible quantum computation for adding arbitrary two numbers
represented in terms of Excess-3 code is investigated. The process of
corresponding arithmetic operation with Adders is illustrated in detail.

Let E_A^B denote the elliptic curve E_A^B:y^2=x^3 Ax B. In this paper, we calculate the number of points on elliptic curves E_A^0:y^2=x^3 Ax over \mathbb{F}_p mod 24. For example, if p{\equiv}1 (mod 24) is a prime, 3t^2{\equiv}1 (mod p) and A(-1 + 2t) is a quartic residue modulo p, then the number of points in E_A^0:y^2=x^3 Ax is congruent to 0 mod...

In this article, we first aim to give simple proofs of known formulae for the generalized Carlitz q-Bernoulli polynomials β
m,χ
(x, q) in the p-adic case by means of a method provided by Kim and then to derive a complex, analytic, two-variable q-L-function that is a q-analog of the two-variable L-function. Using this function, we calculate the valu...

Let . Next, the convolution sums , , etc., are evaluated for all with , .

The q-analogues of many well known formulas are derived by using several results of q-Bernoulli, q-Euler numbers and polynomials. The q-analogues of ζ-type functions are given by using generating functions of q-Bernoulli, q-Euler numbers and polynomials. Finally, their values at non-positive integers are also been computed.
2010 Mathematics Subjec...

Let E: Y2 = 4x3 + Ax + B, with A, B ∈ ℝ be an elliptic curve defined over ℝ. We know that E(C) ≃ C/L for some lattice L. The goal of this paper is to show that L is either rectangular or a special shape of parallelogram and deduce that for g2 (τ), g3(τ) ∈ ℝ, the Weierstrass ℘-function has real number.

In this paper, we propose a new finger biometric method. Infrared finger images are first captured, and then feature extraction is performed using a modified gaussian high-pass filter through binarization, local binary pattern (LBP), and local derivative pattern (LDP) methods. Infrared finger images include the multimodal features of finger veins a...

We deal with an ant colony based routing model for wireless multi‐hop networks. Our model adopts an elliptic curve equation, which is beneficial to design pheromone dynamics for load balancing and packet delivery robustness. Due to the attribute of an elliptic curve equation, our model prevents the over‐utilization of a specific node, distinctively...

Mobility support in 6LoWPAN increases the fault tolerance capacity, connectivity, allows extending and adapting network to changes of location and infrastructure. These features are necessary to satisfy the dependability and scalability of the networks of the future world. Several solutions have been developed to support mobility, but they present...

Future Internet of Things requires mobility support for extending and adapting the network to changes of location and infrastructure, increases the fault tolerance capacity, connectivity, dependability and scalability. The current Future Internet architectures are based on ID/Locator split to support the mobility, but these approaches are not, on t...

Information theories for the general time-dependent harmonic oscillator are described on the basis of invariant operator method. We obtained entropic uncertainty relation of the system and discussed whether it is always larger than or equal to the physically allowed minimum value. Shannon information and Fisher information are derived by means of d...

In this paper, we proposed optimized wireless ECG monitoring system with consideration of various factors from ECG in medical practice and wireless sensor network. For same, we introduced ECG data recording scheme and Wireless sensor network model that is efficient in terms of energy and space requirements without affecting the various parameter of...

Designing low power and less delay for mobile nodes is one of the most important issues for the ubiquitous sensor networks (USN). The paper presents a novel Micro Mobility Sensor Protocol (MMSP), as an enhanced form of the AODV (Ad hoc on-demand Distance Vector) protocol, in order to improve the quality of mobile IP (IPv6)-USN nodes. An IP-USN node...

The paper is devoted to logical implication of MANet’s protocols for gateway discovery in IPv6 Wireless Sensor Networks (IPv6-WSN). We implement three approaches in NS-2 simulator and evaluate its performance under the specified field of the networks of gateway discovery. The performance results present the packet delivery ratio and delay of end to...

We consider q-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds. Finally, we investigate some interesting properties of the modified q-Bernstein polynomials related to q-Euler numbers and
q-Stirling numbers by using fermionic p-adic integrals on .

The main purpose of this paper is to present explicit formulas for the g-analogue of higher order twisted Euler numbers and polynomials using the fermionic p-adic q-integral on ℤp. By using these formulas, we give a relation between the multiple twisted Hurwitz g-zeta function ζq(k)λ(s, x) and the multiple twisted two-variable q-l-function lq(k)λ(s...

We consider some of the properties of divisor functions arising from q-series and theta functions. Using these we obtain several new identities involving divisor functions. On the other hand we prove a conjecture of Z. H. Sun concerning representations by the ternary quadratic form x 2 +y 2 +3z 2 , and also get an analogous result for x 2 +y 2 +2z...

The aim of this paper is to give relations between generalized Dedekind eta functions, theta functions, Dedekind sums, Hardy‐Berndt sums and Hecke operators.

For an infinite family of modular forms constructed from Klein forms we provide certain explicit formulas for their Fourier coefficients by using the theory of basic hypergeometric series (Theorem 2). By making use of these modular forms we investigate the bases of the vector spaces of modular forms of some levels (Theorem 5) and find its applicati...

Main purpose of this paper is to define an elliptic analogue of the Hardy sums. Some results, which are related to elliptic analogue of the Hardy sums, are given.

We first prove Sun's three conjectures [Z.H. Sun, On the number of incongruent residues of x4+ax2+bx modulo p, J. Number Theory 119 (2006) 210–241; Z.H. Sun, http://sfb.hytc.edu.cn/xsjl/szh/, 2000, June] on the number of rational points of some elliptic curves over finite fields Fp, which are related to the congruence cubic and quartic residue. And...

The main purpose of this paper is to study on generating functions of the q-Genocchi numbers and polynomials. We prove new relation for the generalized q-Genocchi numbers which is related to the q-Genocchi numbers and q-Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define q-Genocchi zet...

In this paper, by using q-deformed bosonic p-adic integral, we give λ-Bernoulli numbers and polynomials, we prove Witt's type for-mula of λ-Bernoulli polynomials and Gauss multiplicative formula for λ-Bernoulli polynomials. By using derivative operator to the generating functions of λ-Bernoulli polynomials and generalized λ-Bernoulli num-bers, we g...

Let k be an imaginary quadratic field, h the complex upper half plane, and let τ ∈ h∩k, q = e πiτ . We find a lot of algebraic properties derived from theta functions, and by using this we explore some new algebraic numbers from Rogers-Ramanujan continued fractions.

In this paper, we investigate some relations between Bernoulli numbers and Frobenius-Euler numbers, and we study the values for p-adic l-function.

In this paper, by using q-Volkenborn integral[10], the first author[25] constructed new generating functions of the new twisted (h, q)-Bernoulli polynomials and numbers. We define higher-order twisted (h, q)-Bernoulli polynomials and numbers. Using these numbers and polynomials, we obtain new approach to the complete sums of products of twisted (h,...

This paper relates succinctly how Hardy’s concept of summation can be associated with the Lerch zeta function that leads to the Bose-Einstein integral, and ultimately to the Lindelöf-Wirtinger analytic expansion. By using the generating function of q-Bernoulli numbers, we also define the generating function of the Bose-Einstein type kernel.

In this study, we construct the two-variable Dirichlet q-L-function and the two-variable multiple Dirichlet-type Changhee q-L-function. These functions interpolate the q-Bernoulli polynomials and generalized Changhee q-Bernoulli polynomials. By using the Mellin transformation, we give an integral representation for the two-variable multiple Dirichl...

In this paper, we calculate the number of points on elliptic curves y^2 = x^3 + cx over F_p mod 8.