Euler Polynomials Of Higher Order Research Articles

Derivation of computational formulas for certain class of finite sums: Approach to generating functions arising from p$$ p $$‐adic integrals and special functions

The aim of this paper is to construct generating functions for certain families of special finite sums by using the Newton–Mercator series, hypergeometric functions, and ‐adic integral. By using these generating functions with their functional and partial derivative equations, many novel computational formulas involving the special finite sums of (inverse) binomial coefficients, the Bernoulli type polynomials and numbers, Euler polynomials and numbers, the Stirling numbers, the (alternating) harmonic numbers, the Leibnitz polynomials, and others are derived. We also develop a computation algorithm for these finite sums and provide some of their special values. By using these finite sums and combinatorial numbers, we find some formulas involving multiple alternating zeta functions, the Bernoulli polynomials of higher order and the Euler polynomials of higher order. We then obtain a decomposition from these formulas, which are related to the multiple Hurwitz zeta functions.

On the type 2 poly-Bernoulli polynomials associated with umbral calculus

Abstract Type 2 poly-Bernoulli polynomials were introduced recently with the help of modified polyexponential functions. In this paper, we investigate several properties and identities associated with those polynomials arising from umbral calculus techniques. In particular, we express the type 2 poly-Bernoulli polynomials in terms of several special polynomials, like higher-order Cauchy polynomials, higher-order Euler polynomials, and higher-order Frobenius-Euler polynomials.

Note on the Type 2 Degenerate Multi-Poly-Euler Polynomials

Kim and Kim (Russ. J. Math. Phys. 26, 2019, 40-49) introduced polyexponential function as an inverse to the polylogarithm function and by this, constructed a new type poly-Bernoulli polynomials. Recently, by using the polyexponential function, a number of generalizations of some polynomials and numbers have been presented and investigated. Motivated by these researches, in this paper, multi-poly-Euler polynomials are considered utilizing the degenerate multiple polyexponential functions and then, their properties and relations are investigated and studied. That the type 2 degenerate multi-poly-Euler polynomials equal a linear combination of the degenerate Euler polynomials of higher order and the degenerate Stirling numbers of the first kind is proved. Moreover, an addition formula and a derivative formula are derived. Furthermore, in a special case, a correlation between the type 2 degenerate multi-poly-Euler polynomials and degenerate Whitney numbers is shown.

Asymptotics of Genocchi polynomials and higher order Genocchi polynomials using residues

Approximation formulas for the Genocchi polynomials and Genocchi polynomials of higher order are obtained using the method of residues. The authors were motivated by a paper of Lopez and Temme. Formulas obtained here are used as check formulas of their formulas for Euler polynomials and higher order Euler polynomials. Moreover, asymptotic formulas of Genocchi numbers and higher order Genocchi numbers are obtained.

A new approach to Legendre-truncated-exponential-based Sheffer sequences via Riordan arrays⋆

The significance of multi-variable special polynomials has been identified both in mathematical and applied frameworks. The article aims to focus on a new class of 3-variable Legendre-truncated-exponential-based Sheffer sequences and to investigate their properties by means of Riordan array techniques. The quasi-monomiality of these sequences is studied within the context of Riordan arrays. These sequences are expressed in determinant forms by utilizing the relation between the Sheffer sequences and Riordan arrays. Certain special Sheffer sequences are used to construct the Legendre-truncated-exponential-based Chebyshev polynomials. The same polynomials are used as base to introduce the hybrid classes involving the exponential polynomials and the Euler polynomials of higher order as illustrative examples of the aforementioned general class of polynomials. The shapes are shown and zeros are computed for these sequences by using mathematical software. The main purpose is to demonstrate the advantage of using numerical investigation and computations to discover fascinating pattern of scattering of zeros through graphical representations.

Orthogonal polynomials and connection to generalized Motzkin numbers for higher-order Euler polynomials

We study the higher-order Euler polynomials and give the corresponding monic orthogonal polynomials, which are the Meixner–Pollaczek polynomials with certain arguments and constant factors. Moreover, we obtain a connection to the generalized Motzkin number, which leads to a new recurrence formula and a matrix representation for the higher-order Euler polynomials. Analogue on Bernoulli polynomials is presented in the end.

Reciprocal relations of Bernoulli and Euler numbers/polynomials

ABSTRACTBy means of the symmetric summation theorem on polynomial differences due to Chu and Magli [Summation formulae on reciprocal sequences. European J Combin. 2007;28(3):921–930], we examine Bernoulli and Euler polynomials of higher order. Several reciprocal relations on Bernoulli and Euler numbers and polynomials are established, including some recent ones obtained by Agoh Shortened recurrence relations for generalized Bernoulli numbers and polynomials. J Number Theory. 2017;176:149–173.

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.

On a q-analog of some numbers and polynomials

In this paper, we introduce new q-analogs of the Changhee numbers and polynomials of the first kind and of the second kind. We also derive some new interesting identities related to the Stirling numbers of the first kind and of the second kind, the Euler polynomials of higher order and the q-analogs of Euler polynomials by applying the p-adic integrals method and some summation transform techniques. It turns out that some well-known results are derived as special cases.

On the sum of products of the second kind (h,q)-Euler polynomials of higher order

On the sum of products of the second kind (h,q)-Euler polynomials of higher order

THEOREMS ON GENOCCHI POLYNOMIALS OF HIGHER ORDER ARISING FROM GENOCCHI BASIS

Recently, Kim \textit{et al.} [8] constructed a new method to obtain interesting identities related to Euler polynomials of higher order arising from Euler basis. In the present paper, we study to Genocchi polynomials of higher order arising from Genocchi basis by using the method of Kim \textit{et al}. We also derive many interesting properties related to Genocchi polynomials of higher order.

Identities of symmetry for (h,q)−extension of higher-order Euler polynomials

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, which is an answer to a part of open questions in [7]. This paper is also shown in arXiv( see

Some Identities Involving Euler Polynomials Arising from a Non-linear Differential Equation

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.

Some identities of higher order Euler polynomials arising from Euler basis

The purpose of this paper is to present a systematic study of some families 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 polynomials.

Genocchi polynomials associated with the Umbral algebra

Abstract The aim of this paper is to study on the Genocchi polynomials of higher order on P, the algebra of polynomials in the single variable x over the field C of characteristic zero and P ∗ , the vector spaces of all linear functional on P. By using the action of a linear functional L on a polynomial p ( x ) Sheffer sequences and Appell sequences, we obtain some fundamental properties of the Genocchi polynomials. Furthermore, we give relations between, the first and second kind Stirling numbers, Euler polynomials of higher order and Genocchi polynomials of higher order.

Identities of symmetry for higher-order Euler polynomials in three variables (II)

We derive twenty five basic identities of symmetry in three variables related to higher-order Euler polynomials and alternating power sums. This demonstrates that there are abundant identities of symmetry in three-variable case, in contrast to two-variable case, where there are only a few. These are all new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the p-adic integral expression of the generating function for the higher-order Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating power sums.

Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind

Recently, the authors introduced some generalizations of the Apostol–Bernoulli polynomials and the Apostol–Euler polynomials (see [Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290–302] and [Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917–925]). The main object of this paper is to investigate an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol–Genocchi polynomials of higher order. For these generalized Apostol–Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol–Bernoulli polynomials and the Apostol–Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper [Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631–642] and pose two open problems on the subject of our investigation.

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.

Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on ℤ p

The objective of the paper is to indicate a symmetry of the multivariate p-adic invariant integral on ℤ p , which leads to a relation between the power sum polynomials and higher-order Euler polynomials.

Note on the -Extension of Barnes' Type Multiple Euler Polynomials

We construct the -Euler numbers and polynomials of higher order, which are related to Barnes' type multiple Euler polynomials. We also derive many properties and formulae for our -Euler polynomials of higher order by using the multiple integral equations on .