scholarly journals Explicit formulas for Euler polynomials and Bernoulli numbers

2021 ◽  
Vol 27 (4) ◽  
pp. 80-89
Author(s):  
Laala Khaldi ◽  
◽  
Farid Bencherif ◽  
Miloud Mihoubi ◽  
◽  
...  
Keyword(s):  
Linear Combination ◽  
Bernoulli Numbers ◽  
Euler Polynomial ◽  
Euler Polynomials ◽  
Explicit Formulas ◽  
Fixed Integer

In this paper, we give several explicit formulas involving the n-th Euler polynomial E_{n}\left(x\right). For any fixed integer m\geq n, the obtained formulas follow by proving that E_{n}\left(x\right) can be written as a linear combination of the polynomials x^{n}, \left(x+r\right)^{n},\ldots, \left(x+rm\right)^{n}, with r\in \left \{1,-1,\frac{1}{2}\right\}. As consequence, some explicit formulas for Bernoulli numbers may be deduced.

scholarly journals Notes on the Poly-Korobov polynomials and related polynomials

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 469-474
Author(s):  
Burak Kurt
Keyword(s):  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Polynomials ◽  
Changhee Polynomials

In recent years, many mathematicians ([2], [7], [8], [9], [15], [16], [21]) introduced and investigated for the Korobov polynomials. They gave some identities and relations for the Korobov type polynomials. In this work, we give some relations for the first kind Korobov polynomials and Korobov type Changhee polynomials. Further, wegive two relations between the poly-Changhee polynomials and the poly-Korobov polynomials. Also, we give a relation among the poly-Korobov type Changhee polynomials, the Stirling numbers of the second kind, the Euler polynomials and the Bernoulli numbers.

scholarly journals Old and New Identities for Bernoulli Polynomials via Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Luis M. Navas ◽  
Francisco J. Ruiz ◽  
Juan L. Varona
Keyword(s):  
Fourier Series ◽  
Linear Combination ◽  
Legendre Polynomials ◽  
Fourier Coefficients ◽  
Bernoulli Polynomials ◽  
Bernoulli Numbers ◽  
Gegenbauer Polynomials ◽  
Linear Combinations ◽  
The Given

The Bernoulli polynomialsBkrestricted to[0,1)and extended by periodicity haventh sine and cosine Fourier coefficients of the formCk/nk. In general, the Fourier coefficients of any polynomial restricted to[0,1)are linear combinations of terms of the form1/nk. If we can make this linear combination explicit for a specific family of polynomials, then by uniqueness of Fourier series, we get a relation between the given family and the Bernoulli polynomials. Using this idea, we give new and simpler proofs of some known identities involving Bernoulli, Euler, and Legendre polynomials. The method can also be applied to certain families of Gegenbauer polynomials. As a result, we obtain new identities for Bernoulli polynomials and Bernoulli numbers.

scholarly journals Explicit inverse of the Pascal matrix plus one

2006 ◽  
Vol 2006 ◽  
pp. 1-7 ◽  
Author(s):  
Sheng-liang Yang ◽  
Zhong-kui Liu
Keyword(s):  
Bernoulli Numbers ◽  
Simple Approach ◽  
Euler Polynomials ◽  
Pascal Matrix ◽  
The Matrix

This paper presents a simple approach to invert the matrixPn+Inby applying the Euler polynomials and Bernoulli numbers, wherePnis the Pascal matrix.

scholarly journals New Generalized Apostol-Frobenius-Euler polynomials and their Matrix Approach

2021 ◽  
Vol 45 (03) ◽  
pp. 393-407
Author(s):  
MARÍA JOSÉ ORTEGA ◽  
WILLIAM RAMÍREZ ◽  
ALEJANDRO URIELES
Keyword(s):  
Polynomial Matrix ◽  
Product Formula ◽  
Euler Polynomial ◽  
Euler Polynomials ◽  
Matrix Approach ◽  
Pascal Matrix ◽  
Differential Properties ◽  
Euler Matrix

In this paper, we introduce a new extension of the generalized Apostol-Frobenius-Euler polynomials ℋn[m−1,α](x; c,a; λ; u). We give some algebraic and differential properties, as well as, relationships between this polynomials class with other polynomials and numbers. We also, introduce the generalized Apostol-Frobenius-Euler polynomials matrix ????[m−1,α](x; c,a; λ; u) and the new generalized Apostol-Frobenius-Euler matrix ????[m−1,α](c,a; λ; u), we deduce a product formula for ????[m−1,α](x; c,a; λ; u) and provide some factorizations of the Apostol-Frobenius-Euler polynomial matrix ????[m−1,α](x; c,a; λ; u), which involving the generalized Pascal matrix.

Some Explicit Formulas for the Frobenius-Euler Polynomials of Higher Order

2017 ◽  
Vol 11 (2) ◽  
pp. 621-626 ◽  
Author(s):  
H. M. Srivastava ◽  
Mohamed Amine Boutiche ◽  
Mourad Rahmani
Keyword(s):  
Higher Order ◽  
Euler Polynomials ◽  
Explicit Formulas

scholarly journals Some Identities of Degenerate Bell Polynomials

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 40 ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Jongkyum Kwon
Keyword(s):  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Bell Polynomials ◽  
Euler Polynomials ◽  
New Type ◽  
Degenerate Version

The new type degenerate of Bell polynomials and numbers were recently introduced, which are a degenerate version of Bell polynomials and numbers and are different from the previously introduced partially degenerate Bell polynomials and numbers. Several expressions and identities on those polynomials and numbers were obtained. In this paper, as a further investigation of the new type degenerate Bell polynomials, we derive several identities involving those degenerate Bell polynomials, Stirling numbers of the second kind and Carlitz’s degenerate Bernoulli or degenerate Euler polynomials. In addition, we obtain an identity connecting the degenerate Bell polynomials, Cauchy polynomials, Bernoulli numbers, Stirling numbers of the second kind and degenerate Stirling numbers of the second kind.

scholarly journals Explicit Formulas Involving -Euler Numbers and Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Serkan Araci ◽  
Mehmet Acikgoz ◽  
Jong Jin Seo
Keyword(s):  
Generating Function ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Explicit Formulas ◽  
Derivative Operator ◽  
Euler Numbers And Polynomials ◽  
Integer Ring

We deal with -Euler numbers and -Bernoulli numbers. We derive some interesting relations for -Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the -Euler numbers and -Bernoulli numbers via the -adic -integral in the -adic integer ring.

scholarly journals Truncated euler polynomials

2018 ◽  
Vol 68 (3) ◽  
pp. 527-536 ◽  
Author(s):  
Takao Komatsu ◽  
Claudio Pita-Ruiz
Keyword(s):  
Bernoulli Polynomial ◽  
Euler Polynomial ◽  
Euler Polynomials

Abstract We define a truncated Euler polynomial Em,n(x) as a generalization of the classical Euler polynomial En(x). In this paper we give its some properties and relations with the hypergeometric Bernoulli polynomial.

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