Generalized Euler Polynomials Research Articles

On [formula omitted]th order Euler polynomials of degree [formula omitted] that are Eisenstein

For m an even positive integer and p an odd prime, we show that the generalized Euler polynomial Emp(mp)(x) is in Eisenstein form with respect to p if and only if p does not divide m(2m−1)Bm. As a consequence, we deduce that at least 1/3 of the generalized Euler polynomials En(n)(x) are in Eisenstein form with respect to a prime p dividing n and, hence, irreducible over Q.

New recurrence relations for several classical families of polynomials

In this paper, we derive new recurrence relations for the following families of polynomials: Nørlund polynomials, generalized Bernoulli polynomials, generalized Euler polynomials, Bernoulli polynomials of the second kind, Buchholz polynomials, generalized Bessel polynomials and generalized Apostol–Euler polynomials. The recurrence relations are derived from a differential equation of first order and a Cauchy integral representation obtained from the generating function of these polynomials.

A second type of higher order generalized geometric polynomials and higher order generalized Euler polynomials

In this study we introduce a second type of higher order generalized geometric polynomials. This we achieve by examining the generalized stirling numbers S(n, k, α, β, γ) [Hsu and Shiue, 1998] for some negative arguments. We study their number theoretic properties, asymptotic properties, and their combinatorial properties using the notion of barred preferential arrangements. We also proposed a generalisation of the classical Euler polynomials and show how these generalized Euler polynomials are related to the second type of higher order generalized geometric polynomials.

Numerator polynomials of Riordan matrices

Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. Generalized Euler polynomials gn(x)=(1−x)n+1∑m=0∞pn(m)xm, where pn(m) is the polynomial of degree ≤n, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan matrices. Generalized Narayana polynomials hn(x)=(1−x)2n+1∑m=0∞(m+1)...(m+n)pn(m)xm are the numerator polynomials of the generating functions of diagonals of the exponential Riordan matrices. In paper, the properties of these two types of numerator polynomials and the constructive relationships between them are considered.

Explicit Formulas Associated with Some Families of Generalized Bernoulli and Euler Polynomials

In this paper, we propose and derive several new explicit formulas of the generalized Bernoulli and Euler polynomials in terms of the generalized Stirling numbers of the second kind. A study of some families of the modified generalized Euler polynomials yields an interesting algorithm for calculating the generalized Euler polynomials.

Symmetric properties for generalized Euler polynomials of the second kind

Symmetric properties for generalized Euler polynomials of the second kind

Limiting zeros of sampled systems with time delay

We consider the problem of asymptotic analysis of the zeros of a sampled system of a linear time-invariant continuous system as the sampling period decreases. We show that for a continuous prototype system with delay the limits of a part of the model's zeros are roots of certain polynomials whose coefficients are determined by the relative order of the prototype system and delay. In the special case of zero delay these polynomials coincide with Euler polynomials. Zeros of these generalized Euler polynomials are localized: we show that they are all simple and negative, and that they move monotonically between the zeros of classical Euler polynomials as the fractional part of the delay divided by the sampling period grows. Our results lead to sufficient and "almost necessary" conditions for the stable invertible of the discrete model for all sufficiently small values of the sampling period.

Identities for generalized Euler polynomials

For N∈ℕ, let TN be the Chebyshev polynomial of the first kind. Expressions for the sequence of numbers , defined as the coefficients in the expansion of 1/TN(1/z), are provided. These coefficients give formulas for the classical Euler polynomials in terms of the so-called generalized Euler polynomials. The proofs are based on a probabilistic interpretation of the generalized Euler polynomials recently given by Klebanov et al. Asymptotics of are also provided.

AN EXTENSION OF GENERALIZED EULER POLYNOMIALS OF THE SECOND KIND

Many mathematicians have studied various relations beween Euler number <TEX>$E_n$</TEX>, Bernoulli number <TEX>$B_n$</TEX> and Genocchi number <TEX>$G_n$</TEX> (see [1-18]). They have found numerous important applications in number theory. Howard, T.Agoh, S.-H.Rim have studied Genocchi numbers, Bernoulli numbers, Euler numbers and polynomials of these numbers [1,5,9,15]. T.Kim, M.Cenkci, C.S.Ryoo, L. Jang have studied the q-extension of Euler and Genocchi numbers and polynomials [6,8,10,11,14,17]. In this paper, our aim is introducing and investigating an extension term of generalized Euler polynomials. We also obtain some identities and relations involving the Euler numbers and the Euler polynomials, the Genocchi numbers and Genocchi polynomials.

ON THE GENERALIZED EULER POLYNOMIALS OF THE SECOND KIND

In this paper, our aim is nding the term of generalized Euler polynomials. We also obtain some identities and relations involving the Bernoulli numbers, the Euler numbers and the Stirling numbers.

Distribution of the zeros of the generalized Euler polynomials of second kind

In this paper, we observe the behavior of complex roots of the generalized Euler polynomials En(x) of the second kind, using numerical investigation. By means of numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the generalized Euler polynomials En(x) of the second kind. Finally, we give a table for the solutions of the generalized Euler polynomials En(x) of the second kind.

Exploring the zeros of the generalized Euler polynomials of second kind

Exploring the zeros of the generalized Euler polynomials of second kind

Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

Various interesting and potentially useful properties and relationships involving the Bernoulli, Euler and Genocchi polynomials have been investigated in the literature rather extensively. Recently, the present authors (Srivastava and Pinter in Appl Math Lett 17:375–380, 2004) obtained addition theorems and other relationships involving the generalized Bernoulli polynomials \({B_n^{(\alpha)}(x)}\) and the generalized Euler polynomials \({E_n^{(\alpha)}(x)}\) of order α and degree n in x. The main purpose of this sequel to some of the aforecited investigations is to give several addition formulas for a general class of Appell sequences. The addition formulas, which are derived in this paper, involve not only the generalized Bernoulli polynomials \({B_n^{(\alpha)}(x)}\) and the generalized Euler polynomials \({E_n^{(\alpha)}(x)}\) , but also the generalized Genocchi polynomials \({G_n^{(\alpha)}(x)}\) , the Srivastava polynomials \({\mathcal{S}_{n}^{N}\left( x\right)}\) , several general families of hypergeometric polynomials and such orthogonal polynomials as the Jacobi, Laguerre and Hermite polynomials. Some umbral-calculus generalizations of the addition formulas are also investigated.

Generalized -Euler Numbers and Polynomials

We generalize the Euler numbers and polynomials by the generalized -Euler numbers and polynomials . For the complement theorem, have interesting different properties from the Euler polynomials and we observe an interesting phenomenon of “scattering” of the zeros of the the generalized Euler polynomials in complex plane.

On the generalization of the Euler polynomials with the real parameters

In this study, we give multiplication formula for generalized Euler polynomials of order α and obtain some explicit recursive formulas. The multiple alternating sums with positive real parameters a and b are evaluated in terms of both generalized Euler and generalized Bernoulli polynomials of order α . Finally we obtained some interesting special cases.

Identities of Symmetry for Generalized Euler Polynomials

We derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the -adic fermionic integral expression of the generating function for the generalized Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating generalized power sums.

The Multiple Hurwitz Zeta Function and the Multiple Hurwitz-Euler Eta Function

Almost eleven decades ago, Barnes introduced and made a systematic investigation on the multiple Gamma functions $\Gamma_n$. In about the middle of 1980s, these multiple Gamma functions were revived in the study of the determinants of Laplacians on the $n$-dimensional unit sphere ${\bf S}^n$ by using the multiple Hurwitz zeta functions $\zeta_n(s,a)$. In this paper, we first aim at presenting a generalized Hurwitz formula for $\zeta_n(s,a)$ together with its various special cases. Secondly, we give analytic continuations of multiple Hurwitz-Euler eta function $\eta_n(s,a)$ in two different ways. As a by-product of our second investigation, a relationship between $\eta_n(-\ell,a)$ $(\ell \in \mathbb{N}_0)$ and the generalized Euler polynomials $E_\ell^{(n)}(n-a)$ is also presented.

A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials

The goal of this paper is to unify and extend the generating functions of the generalized Bernoulli polynomials, the generalized Euler polynomials and the generalized Genocchi polynomials associated with the positive real parameters a and b and the complex parameter β . By using this generating function, we derive recurrence relations and other properties for these polynomials. By applying the Mellin transformation to the generating function of the unification of Bernoulli, Euler and Genocchi polynomials, we construct a unification of the zeta functions. Furthermore, we give many properties and applications involving the functions and polynomials investigated in this paper.

On the Symmetric Properties of the Multivariate p-Adic Invariant Integral on ℤp Associated with the Twisted Generalized Euler Polynomials of Higher Order

We study the symmetric properties for the multivariate -adic invariant integral on related to the twisted generalized Euler polynomials of higher order.

Interpolation function of the (h,q)-extension of twisted Euler numbers

In [H. Ozden, Y. Simsek, I.N. Cangul, Generating functions of the ( h , q ) -extension of Euler polynomials and numbers, Acta Math. Hungarica, in press (doi:10.1007/510474-008-7139-1)], by using p -adic q -invariant integral on Z p in the fermionic sense, Ozden et al. constructed generating functions of the ( h , q ) -extension of Euler polynomials and numbers. They defined ( h , q ) -Euler zeta functions and ( h , q ) -Euler l -functions. They also raised the following problem: “ Find a p - adic twisted interpolation function of the generalized twisted ( h , q ) - Euler numbers, E n , χ , w ( h ) ( q ) ”. The aim of this paper is to give a partial answer to this problem. Therefore, we constructed twisted ( h , q ) -partial zeta function and twisted p -adic ( h , q ) -Euler l -functions: l E , p , ξ , q ( h ) ( s , χ ) = 2 ∑ m = 1 ( m , p ) = 1 ∞ χ ( m ) ( − 1 ) m ξ m q h m m s , which interpolate ( h , q ) -extension of Euler numbers, at negative integers: l E , p , ξ , q ( h ) ( − n , χ ) = E n , ξ , χ n ( h ) ( q ) − p n χ n ( p ) E n , ξ p , χ n ( h ) ( q p ) . By using this interpolation function and twisted ( h , q ) -partial zeta function, we proved distribution relations of the ( h , q ) -extension of generalized Euler polynomials. Consequently we find a partial answer to the above question.