Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus
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RESEARCH
Open Access
Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus Dae San Kim1 , Taekyun Kim2* , Dmitry V Dolgy3 and Seog-Hoon Rim4 *
Correspondence: [email protected] 2 Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea Full list of author information is available at the end of the article
Abstract In this paper, we derive the identities of higher-order Bernoulli, Euler and Frobenius-Euler polynomials from the orthogonality of Hermite polynomials. Finally, we give some interesting and new identities of several special polynomials arising from umbral calculus. MSC: 05A10; 05A19 Keywords: Bernoulli polynomial; Euler polynomial; Abel polynomial
1 Introduction The Hermite polynomials are defined by the generating function to be
ext–t = eH(x)t =
∞
Hn (x)
n=
tn n!
(.)
with the usual convention about replacing H n (x) by Hn (x) (see []). In the special case, x = , Hn () = Hn are called the nth Hermite numbers. From (.) we have Hn (x) = (H + x)n =
n n Hn–l xl l . l
(.)
l=
Thus, by (.), we get n! dk Hn–k (x), Hn (x) = k (n)k Hn–k (x) = k k dx (n – k)!
(.)
where (x)k = x(x – ) · · · (x – k + ). As is well known, the Bernoulli polynomials of order r are defined by the generating function to be
t et –
r ext =
∞ n=
B(r) n (x)
tn n!
(r ∈ R).
(.)
(r) In the special case, x = , B(r) n () = Bn are called the nth Bernoulli numbers of order r (see [–]).
© 2013 Kim et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Kim et al. Advances in Difference Equations 2013, 2013:73 http://www.advancesindifferenceequations.com/content/2013/1/73
Page 2 of 11
The Euler polynomials of order r are also defined by the generating function to be
t e +
r
∞
ext =
En(r) (x)
n=
tn n!
(r ∈ R).
(.)
In the special case, x = , En(r) () = En(r) are called the nth Euler numbers of order r. For λ(= ) ∈ C, the Frobenius-Euler polynomials of order r are given by
–λ et – λ
r ext =
∞
Hn(r) (x|λ)
n=
tn n!
(r ∈ R).
(.)
In the special case, x = , Hn(r) (|λ) = Hn(r) (λ) are called the nth Frobenius-Euler numbers of order r (see [–]). The Stirling numbers of the first kind are defined by the generating function to be
(x)n =
n
S (n, k)xk
(see [, ]),
(.)
k=
and the Stirling numbers of the second kind are given by
et –
n
= n!
∞
S (l, n)
l=n
tl l!
(see []).
(.)
In [] it is known that H (x), H (x), . . . , Hn (x) from an orthogonal basis for the space Pn = p(x) ∈ Q[x]| deg p(x) ≤ n
(.)
with respect to the inner product
p (x), p (x) =
∞
e–x p (x)p (x) dx
(see []).
(.)
–∞
For p(x) ∈ Pn , let us assume that
p(x) =
n
Ck Hk (x).
(.)
k=
Then, from the orthog
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