Mediterr. J. Math.
Springer International Publishing AG,
part of Springer Nature 2018
A New Type of Euler Polynomials and
M. Masjed-Jamei, M. R. Beyki and W. Koepf
Abstract. By deﬁning two speciﬁc exponential generating functions, we
introduce a kind of Euler polynomials and study its basic properties in
detail. As an application of the introduced polynomials, we use them in
computing some new series of Taylor type that contain the associated
Euler numbers E
(0) where E
(x) is the Euler polynomial.
Mathematics Subject Classiﬁcation. 11B68, 11C08, 11Y35.
Keywords. Euler numbers and polynomials, Appell polynomials,
binomial convolution, exponential generating functions.
The Appell polynomials A
(x) deﬁned by
where f is a formal power series in t, have found remarkable applications
in diﬀerent branches of mathematics, theoretical physics and chemistry [1,
2]. One of the special cases of Appell polynomials are Euler polynomials
(x) where E
) are usually known as Euler numbers. These
numbers and polynomials have a close relationship with Bernoulli numbers
and polynomials .
The Euler numbers are deﬁned by an exponential generating function
where, for instance, we have
= 1385, ...
and in general E
=0(n ∈ N).