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.
We construct motivic cohomology classes attached to Rankin--Selberg convolutions of modular forms of weights $\ge 2$, show that these vary analytically in p-adic families, and relate their image under the p-adic regulator map to values of L-functions. As consequences, we prove new cases of Perrin-Riou's conjecture on motivic L-values; we prove finiteness results for Tate--Shafarevich groups for twists of elliptic curves by dihedral Artin characters; and we prove one inclusion in the Iwasawa main conjecture for a single modular form over an imaginary quadratic field.
We compute the Archimedean doubling zeta integrals which appear in the interpolation formulas for the p-adic L-functions of Siegel modular forms, and verify that they agree with the modified Archimedean Euler factors for p-adic interpolation conjectured by Coates--Perrin-Riou.
The main purpose of this paper is to construct not only generating functions of the new approach Genocchi type numbers and polynomials but also interpolation function of these numbers and polynomials which are related to a, b, c arbitrary positive real parameters. We prove multiplication theorem of these polynomials. Furthermore, we give some identities and applications associated with these numbers, polynomials and their interpolation functions.
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
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.
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.
Recently, introduced are the generalized Euler-Genocchi and generalized degenerate Euler-Genocchi polynomials. The aim of this note is to study the multi-Euler-Genocchi and degenerate multi-Euler-Genocchi polynomials which are defined by means of the multiple logarithm and generalize respectively the generalized Euler-Genocchi and generalized degenerate Euler-Genocchi polynomials. Especially, we express the former by the generalized Euler-Genocchi polynomials, the multi-Stirling numbers of the first kind and Stirling numbers of the second kind, and the latter by the generalized degenerate Eule…
Recently, the higher-order q-Euler polynomials and multiple q-Euler zeta functions are introduced by T. Kim ([8, 9]). In this paper, we investigate some symmetric properties of the multiple q-Euler zeta function and derive various identities concerning the higher-order q-Euler polynomials from the symmetric properties of the multiple q-Euler zeta functions.
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.
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.
In the present paper, we introduce Eulerian polynomials attached to by using p-adic q-integral on Zp . Also, we give new interesting identities via the generating functions of Dirichlet's type of Eulerian polynomials. After, by applying Mellin transformation to this generating function of Dirichlet' type of Eulerian polynomials, we derive L-function for Eulerian polynomials which interpolates of Dirichlet's type of Eulerian polynomials at negative integers.
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.
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
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.
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.
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.
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…
The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of two variables. By using these type polynomials, we derive recurrence formulas and some new interesting identities related to the second kind Stirling numbers and generalized Bernoulli polynomials. Moreover, we give the generating function and interpolation function of these modified q-Bernstein polynomials of two variables and also give the derivatives of these polynomials and their generating function.
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]).
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.
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.
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]).
In the present paper, we effect Dirichlet's type of twisted Eulerian polynomials by using p-adic fermionic q-integral on the p-adic integer ring. Also, we introduce some new interesting identities for them. As a result of them, by using contour integral on the generating function of Dirichlet's type of twisted Eulerian polynomials and so we define twisted Eulerian-L-function which interpolates of Dirichlet's type of Eulerian polynomials at negative integers which we state in this paper.
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.
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.
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.
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.
The purpose of this paper is to construct of the unification q-extension Genocchi polynomials. We give some interesting relations of this type of polynomials. Finally, we derive the q-extensions of Hurwitz-zeta type functions from the Mellin transformation of this generating function which interpolates the unification of q-extension of Genocchi 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.
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…
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.
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 …
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.
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.
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.
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.
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.
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.
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.
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.
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.
In this paper, we study some symmetric properties of the multiple q-Euler zeta function. From these properties, we derive several identities of symmetry for the (h;q)-extension of higher-order Euler polynomials.
