Abstract
In this paper, we study umbral calculus to have alternative ways of obtaining our results. That is, we derive some interesting identities of the higher-order Bernoulli, Euler, and Hermite polynomials arising from umbral calculus to have alternative ways.
Highlights
As is well known, the Hermite polynomials are defined by the generating function to be e xt–t = eH(x)t = ∞ tn Hn(x) n!, n= ( . )with the usual convention about replacing Hn(x) by Hn(x)
1 Introduction As is well known, the Hermite polynomials are defined by the generating function to be e xt–t = eH(x)t =
The Bernoulli polynomials of order r are given by the generating function to be t et
Summary
The Bernoulli polynomials of order r are given by the generating function to be t et – r ext =. The higher-order Euler polynomials are defined by the generating function to be et +. (r ∈ R), and En(r)( ) = En(r) are called the nth Euler numbers of order r (see [ – ]). X = , Hn(r)( |λ) = Hn(r)(λ) are called the nth Frobenius-Euler numbers of order r. Let us assume that P is the algebra of polynomials in the variable x over C and that P∗ is the vector space of all linear functionals on P. L|p(x) denotes the action of the linear functional L on a polynomial p(x), and we remind that the vector space structure on P∗ is defined by.
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