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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 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.

SOME q-BERNOULLI NUMBERS OF HIGHER ORDER ASSOCIATED WITH THE p-ADIC q-INTEGRALS

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).

A matrix A =[ a ij ] is called a 7-matrix if its entries satisfy the recurrence relation αa i −1, j −1 + βa i −1, j = a ij where α , β are fixed numbers. A 7-matrix is completely determined by its first row and first column. In this paper we determine the structure of 7-matrices and investigate the sequences represented by columns of infinite 7-matrices.

In this paper we investigate the values of permanents according to the positions of zeros over m × n matrices with k zeros. We determine the maximum and minimum values of the permanent function and find the matrices which attain these extreme values.

In this paper, besides the definition of maximal operators over the p-adic number field using the properties of Section 1, we can define p-adic bounded mean oscillation space and obtain its examples. We prove sufficient conditions of boundedness of maximal operators over the p-adic number field.

By using multiple p-adic q-integrals, we define the p-adic q-L-function in n-variables and the q-extension of p-adic log multiple gamma functions. From these definitions, we show that the values of the p-adic q-L-function at positive integers can be expressed in terms of the q-extension of p-adic log multiple gamma functions.

In this article we characterize Gorenstein rings of Krull dimension at most one by using divisible envelopes. Some of these results were known in [6], where the given ring is an integral domain. We prove these results hold over commutative rings which may not be domains. The motivation of the present discussion is partially from [1], where modules over Gorenstein rings of Krull dimension at most one were studied (see also [[14], Theorem 5.19] for a completely different homological characterization of these rings). We also show that every module over a commutative ring R has a divisible envelope when every S-torsion free R-module has a divisible envelope which is S-torsion free, where S is the set of nonzero-divisors of R.

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.

(2000). On classical artinian localizations. Communications in Algebra: Vol. 28, No. 9, pp. 4455-4462.

In this paper, by using the orthogonality type as defined in the umbral calculus, we derive an explicit formula for several well-known polynomials as a linear combination of the Apostol-Euler polynomials.

In this paper, we investigate some properties of several Sheffer sequences of several polynomials arising from umbral calculus. From our investigation, we can derive many interesting identities of several polynomials, Comment: 10 pages

In this paper, we derive the identities of higher-order Bernoulli, Euler and Frobenius-Euler polynomials from the orthogonality of Hermite polynomials. Finally, we give some interesting and new identities of several special polynomials arising from umbral calculus. MSC: 05A10, 05A19.

In this paper, we investigate some properties of polynomials related to Sheffer sequences. Finally, we derive some identities of higher-order Frobenius-Euler polynomials.

In this paper, we give some interesting identities which are derived from the basis of Genocchi. From our methods which are treated in this paper, we can derive some new identities associated with Bernoulli and Euler polynomials.

In this paper, we derive some identities of Bernoulli, Euler, and Abel polynomials arising from umbral calculus.

We give the analogue of the Lebesgue-Radon-Nikodym theorem with respect to a weighted p-adic q-measure on Z p . In a special case, when the weight q x is 1, we can derive the same result as Kim et al. (Abstr. Appl. Anal. 2011:637634, 2011). And if q = 1 , we have the same result as Kim (Russ. J. Math. Phys. 19:193-196, 2012). MSC:11B68, 11S80.

We will define the extension ofq-Hurwitz zeta function due to Kim and Rim (2000) and study its properties. Finally, we lead to a useful new integral representation for theq-zeta function.

In this paper, we study some p-adic Frobenius-Euler measure related to umbral calculus in the p-adic case. Finally, we derive some identities of Frobenius-Euler polynomials from our study.

In this paper, we derive some interesting identities on Bernoulli and Euler polynomials by using the orthogonal property of Laguerre polynomials.

The purpose of this paper is to investigate some interesting identities on the Bernoulli and Euler polynomials arising from the orthogonality of Legendre polynomials in the inner product space Pn.

In this paper we derive some interesting identities arising from the orhtogonality of gegenbauer polynomials., 11 pages

The purpose of this paper is to give some properties of the modifiedq-Bernoulli numbers and polynomials of higher order with weight. In particular, by using the bosonicp-adicq-integral onℤp, we derive new identities ofq-Bernoulli numbers and polynomials with weight.

We investigate some formulae for the product of two Bernoulli and Euler polynomials arising from the Euler and Bernoulli basis polynomials.

This paper performs a further investigation on the q-Bernoulli polynomials and numbers given by Acikgoz et al. (Adv. Differ. Equ. 2010, 9, Article ID 951764) some incorrect properties are revised. It is pointed out that the definition concerning the q-Bernoulli polynomials and numbers is unreasonable. The purpose of this paper is to redefine the q-Bernoulli polynomials and numbers and correct its wrong properties and rebuild its theorems.

In this paper we investigate some properties for the q-Euler numbers ans polymials. From these properties we give some identities on the Bernstein polymials and q-Euler polynpmials., Comment: 8 pages

A systemic study of some families of 𝑞 -Genocchi numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic 𝑝 -adic integral on ℤ 𝑝 . The study of these higher-order 𝑞 -Genocchi numbers and polynomials yields an interesting 𝑞 -analog of identities for Stirling numbers.

We give a new construction of the -extensions of Euler numbers and polynomials. We present new generating functions which are related to the -Euler numbers and polynomials. We also consider the generalized -Euler polynomials attached to Dirichlet's character and have the generating functions of them. We obtain distribution relations for the -Euler polynomials and have some identities involving -Euler numbers and polynomials. Finally, we derive the -extensions of zeta functions from the Mellin transformation of these generating functions, which interpolate the -Euler polynomials at negative integers.