A note on the generalized Euler numbers and polynomials

A note on a curious formula for Euler's constant

In this short note we will use the residue theorem to establish a formula for Euler's constant. In particular, we offer a slightly generalized version of an interesting infinite series due to Flajolet, Gourdon, and Dumas.

A note on p-adic q-integrals associated with q-Euler numbers

In this we give a detailed proof of fermionic p-adic q-measures on Z_p and we will treat some interesting formulae related q-extension of Euler numbers and polynomials.

A note on degenerate Euler and Bernoulli polynomials of complex variable

In this paper, we study the degenerate version of the new type Euler polynomials, namely degenerate cosine-Euler polynomials and sime-Euler polynomials and also corresponding ones for Bernoulli polynomials, namely degenerate cosine Bernoulli polynomials and degenerate sine-Bernoulli polynomials by considering the degenerate Euler polynomials of complex variable and the degenerate Bernoulli polynomials of complex variable.

Note on the Euler Numbers and Polynomials

In this paper we investigate the properties of the Euler functions. By using the Fourier transform for the Euler function, we derive the interesting formula related to the infinite series. Finally we give some interesting identities between the Euler numbers and the second kind stirling numbers.

A note on the new q-extension of Frobenius-Euler numbers and polynomials

In this paper, we consider the new q-extension of Frobenius-Euler numbers and polynomials and we derive some interesting identities from the orthogonality type properties for the new q-extension of Frobenius-Euler polynomials. Finally we suggest one open question related to orthogonality of polynomials.

Note on q-Dedekind type sums related to q-Euler polynomials

it is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of higher order Dedekind type sums related to q-Euler polynomials and numbers.

A Note On Multi Poly-Euler Numbers And Bernoulli Polynomials

In this paper we introduce the generalization of Multi Poly-Euler polynomials and we investigate some relationship involving Multi Poly-Euler polynomials. Obtaining a closed formula for generalization of Multi Poly-Euler numbers therefore seems to be a natural and important problem.

Higher-order Bernoulli, Frobenius-Euler and Euler polynomials

In this paper, we give some interesting identities of higher-order Bernoulli, Frobenius-Euler and Euler polynomials arising from umbral calculus. From our method of this paper, we can derive many interesting identities of special polynomials

Notes on the module of Euler systems

In this paper, we study the module of Euler systems. We determine the ideal of an Iwasawa algebra associated with Euler systems of rank $0$. We also show that the module of higher rank Euler systems for $\mathbb{G}_{m}$ over a totally real field is free of rank $1$ under the assumptions that Greenberg conjecture holds true and that the $μ$-invariant of a certain Iwasawa module vanishes.

Some identities of higher-order Euler polynomials arising from Euler basis

The purpose of this paper is to present a syatemic study of some familes of higher-order Euler numbers and polynomials. In particular, by using the basis property of higher-order Euler polynomials for the space of polynomials of degree less than and equal to n, we derive some interesting identities for the higher-order Euler polynomias.

广义欧拉多项式的一个恒等式

An Identity for Generalized Euler Polynomials

本文提出广义欧拉多项式的一个新恒等式，并借此推广了欧拉数、欧拉多项式、Genocchi多项式与Genocchi数之间的多类经典关系。

q-Analogue of Euler-Barnes Multiple Zeta Functions (= a note on q-zeta functions)

Recently (see [1]) I has introduced an interesting the Euler-Barnes multiple zeta function. In this paper we construct the q-analogue of Euler-Barnes multiple zeta function which interpolates the q-analogue of Frobenius-Euler numbers of higher order at negative integers.

Note on a sum involving the Euler function

We prove that $$ \sum_{n \leq x} φ([x/n])\leq\bigg(\frac{1380}{4009}+\frac{2629}{4009}\cdot\frac1{ζ(2)}+o(1)\bigg)x\log x $$ as $x\to\infty$, where $φ$ denotes the Euler totient function and $[x]$ denotes the integer part of $x$.

Some identities on Bernstein polynomials associated with q-Euler polynomials

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.

Symmetry properties of the generalized higher-order Euler polynomials

The purpose of this paper is to generalize this relation of symmetry between the power sum polynomials and the generalized Euler polynomials to the relation between the power sum polynomials and the generalized higher-order Euler polynomials.

Note on sums involving the Euler function

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} ϕ\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{ϕ([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. and Wu.

Symmetric identities of q-Euler polynomials

In this paper, we study some symmetric identities of q-Euler numbers and polynomials. From these properties, we derive several identities of q-Euler numbers and polynomials.

Euler number and polynomials of higher order

In this paper we study the higher-order Euler numbers and polynomials and we introduce the mutiple zeta functions which interpolate higher-order Euler polynomials and numbers at negative integers

Truncated Euler polynomials

We define a truncated Euler polynomial $E_{m,n}(x)$ as a generalization of the classical Euler polynomial $E_n(x)$. In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

Some identities of Eulerian polynomials arising from nonlinear differential equations

In this paper, we study nonlinear differential equations arising from Eulerian polynomials and their applications. From our study of nonlinear differential equations, we derive some new and explicit identities involving Eulerian and higher-order Eulerian polynomials.

More Properties on Multi Poly-Euler Polynomials

In this paper, we establish more properties of generalized poly-Euler polynomials with three parameters and we investigate a kind of symmetrized generalization of poly- Euler polynomials. Moreover, we introduce a more general form of multi poly-Euler polynomials and obtain some identities parallel to those of the generalized poly-Euler polynomials.

On the fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials

In this paper, we give a fermionic p-adic integral representions of Benstein polynomials associated with Euler numbers and polynomials. Finally, we give some interesting identities for the Euler numbers by using the properties of our integral represention.

On generalized degenerate Euler-Genocchi polynomials

We introduce the generalized degenerate Euler-Genocchi polynomials as a degenerate version of the Euler-Genocchi polynomials. In addition, we introduce their higher-order version, namely the generalized degenerate Euler-Genocchi polynomials of order α, as a degenerate version of the generalized Euler-Genocchi polynomials of order α. The aim of this paper is to study certain properties and identities involving those polynomials, the generalized falling factorials, the degenerate Euler polynomials of order α, the degenerate Stirling numbers of the second kind, and the alternating degenerate powe…

Some identities of Frobenius-Euler polynomials arising from Frobenius-Euler basis

In this paper, we give some new and interesting identities which are derived from the basis of Frobenius-Euler. Recently, Simsek et als(see [13]) have given some identities of q-analogue of Frobenius-Euler polynomials related to q-Bernstein polynomials. From the methods of our paper, we can also derive the results and identities of Simsek et als (cf.[13]).

Identities involving Frobenius-Euler polynomials arising from non-linear differential equations

In this paper we consider non-linear differential equations which are closely related to the generating functions of Frobenius-Euler polynomials. From our non-linear differential equations, we derive some new identities between the sums of products of Frobenius-Euler polynomials and Frobenius-Euler polynomials of higher order.

On the q-Euler numbers and polynomials with weight 0

The purpose of this paper is to investigate some properties of q-Euler numbers and polynomials with weight 0. From those q-Euler numbers with weight 0, we derive some identities on the q-Euler numbers and polynomials with weight 0.

On the explicit formula of Euler numbers and polynomials of higher order

We study the explicit formula of Euler numbers and polynomials of higher order

Umbral calculus and Euler polynomials

In this paper, we study some properties of Euler polynomials arising from umbral calculus. Finally, we give some interesting identities of Euler polynomials using our results. Recently, Dere and Simsek have studied umbral calculus related to special polynomials (see[6]).

Note on a paper "An Extension of a Theorem of Euler" by Hirata-Kohno et al

In this paper we extend a result of Hirata-Kohno, Laishram, Shorey and Tijdeman on the Diophantine equation $n(n+d)...(n+(k-1)d)=by^2,$ where $n,d,k\geq 2$ and $y$ are positive integers such that $\gcd(n,d)=1.$

Degenrate Eulerian numbers and polynomials

In this paper, we study the degenerate Eulerian polynomials and numbers and give some new and interesting identities associated with several special numbers and polynomials.

New identities involving q-Euler polynomials of higher order

In this paper we give new identities involving q-Euler polynomials of higher order.

Hermite polynomials related to Genocchi, Euler and Bernstein polynomials

The objective of this paper is to derive some interesting properties of Genocchi, Euler and Bernstein polynomials by means of the orthogonality of Hermite polynomials.

Identities involving values of Bernstein, q-Bernoulli and q-Euler polynomials

In this paper we give some relation involving values of q-Bernoulli, q-Euler and Bernstein polynomials. From these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler and Bernstein polynomials.

A finite sum involving generalized falling factorial polynomials and degenerate Eulerian polynomials

The aim of this paper is twofold. Firstly, we investigate a finite sum involving the generalized falling factorial polynomials, in some special cases of which we express it in terms of the degenerate Stirling numbers of the second kind, the degenerate Bernoulli polynomials and the degenerate Frobenius-Euler polynomials. Secondly, we consider the degenerate Eulerian polynomials and deduce the generating function and a recurrence relation for them.

Some identities of q-Euler polynomials arising from q-umbral calculus

Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (See [1]). In this paper, we develop the new method of q-umbral calculus due to Roman and we study new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give some interesting identities on our q-Euler polynomials related to the q-Bernoulli numbers and polynomials of Hegazi and Mansour.

On the q-Euler numbers related to modified q-Bernstein polynomials

In this paper we investigate some interesting formulae of q-Euler numbers and polynomials related to the modified q-Bernstein polynomials.

The Modified q-Euler numbers and polynomials

In the recent paper the interesting q-Euler numbers and polynomials introduced in JMAA. The purpose of this paper is to construct the modified q-Euler numbers and polynomiasl. Finally we will give the interesting many identities related to these numbers and polynomials.

New Generalization of Eulerian polynomials and their applications

In the present paper, we introduce Eulerian polynomials with a and b parameters and give the definition of them. By using the definition of generating function for our polynomials, we derive some new identities in Theory of Analytic Numbers. Also, we give relations between Eulerian polynomials with a and b parameters, Bernstein polynomials, Poly-logarithm function, Bernoulli numbers and Euler numbers. Moreover, we see that our polynomials at a =-1 are related to Euler-Zeta function at negative inetegers. Finally, we get Witt's formula for new generalization of Eulerian polynomials which we exp…

Higher-order frobenius-Euler and poly-Bernoulli mixed type polynomials

In this paper, we consider higher-order Frobenius-Euler polynomi- als associated with poly-Bernoulli polynomials which are derived from polylogarithmic function. These polynomials are called higher-order Frobenius-Euler and poly-Bernoulli mixed type polynomials. The purpose of this paper is to give various identities of those polynomials arising from umbral calculus.

On type 2 degenerate Bernoulli and Euler polynomials of complex variable

Recently, Masjed-Jamei-Beyki-Koepf studied the so called new type Euler polynomials without making use of Euler polynomials of complex variable. Here we study degenerate and type 2 versions of these new type Euler polynomials, namely the type 2 degenerate cosine-Euler and type 2 degenerate sine-Euler polynomials and also the corresponding ones for Bernoulli polynomials, namely the type 2 degenerate cosine-Bernoulli and type 2 degenerate sine-Bernoulli polynomials by considering the degenerate Euler and degenerate Bernoulli polynomials of complex variable and by treating the real and imaginary …

Some identities of degenerate Euler polynomials associated with degenerate Bernstein polynomials

In this paper, we investigate some properties and identities for degenerate Euler polynomials in connection with degenerate Bernstein polynomials by means of fermionic p-adic integrals on Zp and generating functions. In addition, we study two variable degenerate Bernstein polynomials and the degenerate Bernstein operators.

New approach to q-Genocch, Euler numbers and polynomials and their interpolation functions

We give a new construction of q-Genocchi numbers, Euler numbers of higher order, which are different than the q-Genocchi numbers of Cangul-Ozden-Simsek. By using our q-Genoucchi, Euler nimbers of higher order, we can investigate the interesting relationship between w-q-Euler polynomials and w-q-Genocchi polynomials.

Complete sums of products of (h,q)-extension of Euler numbers and polynomials

In this paper we investigate some interesting of the (h,q)-extension of Euler numbers and polynomials. Finally, we will give some relations between these numbers anf polynomials

Some identities of symmetry for the generalized q-Euler polynomials

By the symmetric properties of Drichlet's type multiple q-l-function, we establish various identities concerning the generalized higher-order q-Euler polynomials. Furthermore, we give some interesting relationship between the power sums and the generalized higher-order $q$-Euler polynomials.

On the degenerate Forbenius-Euler polynomials

In this paper, we consider the degenerate Frobenius-Euler polynomials and investigate some identities of these polynomials.

On the q-extension of higher-order Euler polynomials

Thw purpose of this paper is to present a systemic study of some families of the generalized q-Euler numbers and polynomials of higher order.

A $(p,q)$-Analogue of Poly-Euler Polynomials and Some Related Polynomials

In the present article, we introduce a $(p,q)$-analogue of the poly-Euler polynomials and numbers by using the $(p,q)$-polylogarithm function. These new sequences are generalizations of the poly-Euler numbers and polynomials. We give several combinatorial identities and properties of these new polynomials. Moreover, we show some relations with the $(p,q)$-poly-Bernoulli polynomials and $(p,q)$-poly-Cauchy polynomials. The $(p,q)$-analogues generalize the well-known concept of the $q$-analogue.

Apostol-Euler polynomials arising from umbral calculus

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

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.

Representation by degenerate Frobenius-Euler polynomials

The aim of this paper is to represent any polynomial in terms of the degenerate Frobenius-Euler polynomials and more generally of the higher-order degenerate Frobenius-Euler polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

q-Euler numbers and polynomials associated with multiple q-zeta functions

The purpose this paper is to present a systemic study of some families of multiple q-Euler numbers and polynomials and we construct multiple q-zeta function which interpolates multiple q-Euler numbers at negative integers.

Note on q-extensions of Euler numbers and polynomials of higher order

In [14] Ozden-Simsek-Cangul constructed generating functions of higher-order twisted $(h,q)$-extension of Euler polynomials and numbers, by using $p$-adic q-deformed fermionic integral on $\Bbb Z_p$. By applying their generating functions, they derived the complete sums of products of the twisted $(h,q)$-extension of Euler polynomials and numbers, see[13, 14]. In this paper we cosider the new $q$-extension of Euler numbers and polynomials to be different which is treated by Ozden-Simsek-Cangul. From our $q$-Euler numbers and polynomials we derive some interesting identities and we construct $q…

Umbral calculus and Frobenius-Euler polynomials

In this paper, we study some properties of umbral calculus related to Appell sequence. From those properties, we derive new and interesting identities of Frobenius-Euler polynomials.

Analytic Continuation of q-Euler numbers and polynomials

In this paper we study that the $q$-Euler numbers and polynomials are analytically continued to $E_q(s)$. A new formula for the Euler's $q$-Zeta function $ζ_{E,q}(s)$ in terms of nested series of $ζ_{E,q}(n)$ is derived. Finally we introduce the new concept of the dynamics of analytically continued $q$-Euler numbers and polynomials.