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We consider the hyper-geometric Daehee numbrers and polynomials and investigate some properties of those numbers and polynomials.

In this paper, we present differential equation for the generating function of the p, q-Touchard polynomials. An application to ordered partitions of a set is investigated.

In this paper we consider Changhee polynomials of type two, which are motivated from the recent work of Kim and Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmtery for the fermionic p-adic integral on Zp.

We consider Witt-type formula for the extension of Changchee and Daehee numbers and polynomials. We derive some identities and properties of those numbers and polynomials which are related to special polynomials.

The Changhee polynomials and numbers are introduced in (6). Some interesting identities and properties of those polynomials are derived from umbral calculus (see (6)). In this paper, we consider Witt- type formula for the n-th twisted Changhee numbers and polynomials and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to special polynomials.

The Changhee polynomials and numbers are introduced by D.S. Kim et al in [5]. Some interesting identities and properties of those polynomials are derived from umbral calculus(see [5]). In this paper, we consider Witt-type formula for the weighted Changhee numbers and polynomials and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to the weighted Euler polynomials.

In this paper, we construct new q-extension of Euler polynomials with weight 0. These modified q-Euler polynomials are useful to study various identities of Carlitz's q-Bernoulli numbers.

In this paper, we introduce some formulae of series expansion which are derived from trigonometric functions related to Euler and Bernoulli numbers.

In this paper, we consider the q-extensions of Changhee polynomials which are derived from the fermionic p-adic integral on Zp and the q-analogue of falling factorial sequence. Finally, we give some identities of those polynomials.

Recently, Changhee polynomials and numbers are introduced in (6). Some interesting identities and properties of those polynomials are derived from umbral calculus(see (6)). In this paper, we consider Witt-type formula for the Changhee numbers and polynomials and derive some new interesting identities and properties of those polynomials and numbers from the Witt-type formula which are related to special polynomials.

We derive a family of non-linear differential equations from the generating functions of the Euler polynomials and study the solutions of these differential equations. Then we give some new and interesting identities and formulas for the Euler polynomials of higher order by using our non-linear differential equations.

In this paper, we consider Changhee polynomials of type two, which are motivated from the recent work of D. Kim and T. Kim. We investigate some symmetry identities for the Changhee polynomials of type two which are derived from the properties of symmetry for the fermionic p-adic integral on Z p .

In this paper, we construct new q-extension of Bernoulli poly- nomials. These q-Bernoulli polynomials are useful to study various identities of Carlitz's q-Bernoulli numbers.

In this paper, we introduce a new family of functions, which is called the Changhee-Genocchi polynomials. We study some explicit identities on these polynomials, which are related to Genocchi polynomials and Changhee polynomials. Also, we represent Changhee-Genocchi polynomials by gamma and beta functions. We also study some properties of higher-order Changhee-Genocchi polynomials related to Changhee polynomials and Daehee polynomials.

We will give a new proof of the Lebesgue-Radon-Nikodym theorem with respect to weighted p-adic q-measure on , using Mahler expansion of continuous functions, studied by the authors in 2012. In the special case, q = 1, we can derive the same result as in Kim, 2012, Kim et al, 2011.

The main purpose of this paper is to introduce a new type of q-Euler numbers and polynomials with weak weight (a,w): Eawnq`` and Eawnq (x), respectively. By using the fermionic p-adic q-integral on Zp, we can obtain some results and derive some recurrence identities for the q-Euler numbers and polynomials with weak weight (a,w).

We derive some interesting identities and arithmetic properties of Bernoulli and Euler polynomials from the orthogonality of Hermite polynomials. LetPn={p(x)∈ℚ[x]∣deg p(x)≤n}be the(n+1)-dimensional vector space overℚ. Then we show that{H0(x),H1(x),…,Hn(x)}is a good basis for the spacePnfor our purpose of arithmetical and combinatorial applications.

We will give the Lebesgue-Radon-Nikodym theorem with respect to weighted $p$ -adic $q$ -measure on ${\Bbb Z}_{p}$ . In special case, $q=1$ , we can derive the same result as Kim, 2012; Kim et al., 2011.

Recently, some interesting and new identities are introduced in the work of Kim et al. (2012). From these identities, we derive some new and interesting integral formulae for Bernoulli and Genocchi polynomials.

We construct a new type of -Genocchi numbers and polynomials with weight . From these -Genocchi numbers and polynomials with weight , we establish some interesting identities and relations.

We give some new identities on the Bernoulli and Euler numbers by using the bosonicp-adic integral onZpand reflection symmetric properties of Bernoulli and Euler polynomials.

In this paper, we define new kind of Daehee numbers, the degenerate Daehee numbers of the third kind, using degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.

In 2009, Kim et al. gave some identities of symmetry for the twisted Euler polynomials of higher-order, recently. In this paper, we extend our result to the higher-order twisted -Euler numbers and polynomials. The purpose of this paper is to establish various identities concerning higher-order twisted -Euler numbers and polynomials by the properties of -adic invariant integral on . Especially, if , we derive the result of Kim et al. (2009).

We derive the modified -adic -measures related to -Nasybullin's type lemma.

The p -adic invariant q -integral on Z p was originally constructed by T. Kim [T. Kim, On a q -analogue of the p -adic log gamma function and related integrals, J. Number Theory 76 (1999) 320–329]. Recently, many authors have been studying the extended Bernoulli numbers or Euler numbers by using this p -adic q -integral in the fermionic or bosonic sense. Let i ∈ O C p = { x ∈ C p : | x | p ⩽ 1 } . Then we consider new ( i , q ) -Bernoulli and Euler numbers using p -adic q -integrals in this work.

One purpose of this paper is to construct twisted q-Euler numbers by using p-adic invariant integral on Zp in the sense of fermionic. Finally, we consider twisted Euler q-zeta function and q-l-series which interpolate twisted q-Euler numbers and polynomials., 8 pages

Using non-archimedean q-integrals on Z"p defined in [T. Kim, On a q-analogue of the p-adic log gamma functions and related integrals, J. Number Theory 76 (1999) 320-329; T. Kim, q-Volkenborn integration, Russ. J. Math. Phys. 9 (2002) 288-299], we define new Changhee q-Euler polynomials and numbers which are different from those of Kim [T. Kim, p-adic q-integrals associated with the Changhee-Barnes' q-Bernoulli polynomials, Integral Transforms Spec. Funct. 15 (2004) 415-420] and Carlitz [L. Carlitz, q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc. 76 (1954) 332-350]. We define generating functions of multiple q-Euler numbers and polynomials. Furthermore we construct a multivariate Hurwitz type zeta function which interpolates the multivariate q-Euler numbers or polynomials at negative integers.

The purpose of this work is to give some identities of q -Euler numbers and polynomials. Finally we construct q -zeta functions which interpolate the q -analogue of Frobenius–Euler numbers at negative integers.

Cauchy polynomials are also called Bernoulli polynomials of the second kind and these polynomials are very important to study mathematical physics. Kim et al. have studied some properties of Bernoulli polynomials of the second kind associated with special polynomials arising from umbral calculus. Kim introduced the degenerate Cauchy numbers and polynomials which are derived from the degenerate function $e^{t}$ . In this paper, we try to degenerate Cauchy numbers and polynomials k-times and investigate some properties of these k-times degenerate Cauchy numbers and polynomials.

In this paper we construct q-Genocchi numbers and polynomials. By using these numbers and polynomials, we inves- tigate the q-analogue of alternating sums of powers of consecutive integers due to Euler.

A generalized net is used to construct a model which describes the process of evaluation of the problems solved by students. The model utilizes the theory of intuitionistic fuzzy sets. The model can be used to simulate some processes, related to estimation of students` background.

Let n, k be positive integers (k > 1), and let S n , q (k) be the sums of the nth powers of positive q-integers up to k − 1: , where [l] q =(ql −1/q−1). Following an idea due to J. Bernoulli, we explore a formula for S n , q (k) as follows: where β i , q are the q-Bernoulli numbers.

A model of an assessment module has been created, using intuitionisric fuzzy estimations, which render account on the knowledge of the trained objects. The final mark is determined on the basis of a set of evaluation units. An opportunity is offered no only for tracing the changes of the parameters of the trainer object, but there is also an opportunity of tracing the status of the already comprehended knowledge, as well as evaluating and changing the training themes and evaluation criteria

For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension

We study that theq-Bernoulli polynomials, which were constructed by Kim, are analytic continued toβs(z). A new formula for theq-Riemann zeta functionζq(s)due to Kim in terms of nested series ofζq(n)is derived. The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an interesting phenomenon of “scattering” of the zeros ofβs(z)is observed. Following the idea ofq-zeta function due to Kim, we are going to use “Mathematica” to explore a formula forζq(n).