Explicit formulas for the Bernoulli and Euler polynomials and numbers

1991 ◽  
Vol 61 (1) ◽  
pp. 175-180 ◽  
Author(s):  
P. G. Todorov
Keyword(s):  
Euler Polynomials ◽  
Explicit Formulas ◽  
Euler Polynomials And Numbers

scholarly journals Apostol Type (p, q)-Frobenius-Euler Polynomials and Numbers

2017 ◽  
Author(s):  
Ugur Duran ◽  
Mehmet Acikgoz
Keyword(s):  
Difference Equations ◽  
Recurrence Relations ◽  
Integral Representations ◽  
Euler Polynomials ◽  
Addition Theorems ◽  
Explicit Formulas ◽  
Euler Polynomials And Numbers ◽  
Appl Math

In the present paper, we introduce (p, q)-extension of Apostol type Frobenius-Euler polynomials and numbers and investigate some basic identities and properties for these polynomials and numbers, including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, explicit formulas and relations for these polynomials and numbers. Moreover, we discover (p, q)-extensions of Carlitz's result [L. Carlitz, Mat. Mag., 32 (1959), 247–260] and Srivastava and Pintér addition theorems in [H. M. Srivastava, A. Pinter, Appl. Math. Lett., 17 (2004), 375–380].

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.

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 A Note on a New Type of Degenerate poly-Euler Numbers and Polynomials

2020 ◽  
Author(s):  
Waseem Khan
Keyword(s):  
Arithmetic Function ◽  
Bernoulli Numbers ◽  
Euler Polynomials ◽  
Euler Numbers ◽  
Polylogarithm Function ◽  
Euler Numbers And Polynomials ◽  
Special Numbers And Polynomials ◽  
Euler Polynomials And Numbers ◽  
New Type ◽  
Logarithm Function

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

scholarly journals On Multi Poly-Genocchi Polynomials with Parameters a, b and c

2020 ◽  
Vol 13 (3) ◽  
pp. 444-458
Author(s):  
Roberto Bagsarsa Corcino ◽  
Mark Laurente ◽  
Mary Ann Ritzell Vega
Keyword(s):  
Recurrence Relations ◽  
Euler Polynomials ◽  
Explicit Formulas ◽  
Multiple Parameters ◽  
Genocchi Numbers ◽  
Genocchi Polynomials

Most identities of Genocchi numbers and polynomials are related to the well-knownBenoulli and Euler polynomials. In this paper, multi poly-Genocchi polynomials withparameters a, b and c are dened by means of multiple parameters polylogarithm. Several properties of these polynomials are established including some recurrence relations and explicit formulas.

scholarly journals Sums of Products of -Euler Polynomials and Numbers

2009 ◽  
Vol 2009 (1) ◽  
pp. 381324 ◽  
Author(s):  
Young-Hee Kim ◽  
Kyung-Won Hwang ◽  
Taekyun Kim
Keyword(s):  
Euler Polynomials ◽  
Euler Polynomials And Numbers ◽  
Sums Of Products
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