- Open Access
Some identities of Bernoulli, Euler and Abel polynomials arising from umbral calculus
© Kim et al.; licensee Springer 2013
- Received: 26 December 2012
- Accepted: 6 January 2013
- Published: 18 January 2013
In this paper, we derive some identities of Bernoulli, Euler, and Abel polynomials arising from umbral calculus.
- Bernoulli polynomial
- Euler polynomial
- Abel polynomial
where is the Kronecker symbol (see ).
By (1.5), we get . The map is a vector space isomorphism from onto ℱ. So, ℱ denotes both the algebra of formal power series in t and the vector space of all linear functionals on ℙ, and so an element of ℱ is thought of as both a formal power series and a linear functional (see [1–3]). We call ℱ the umbral algebra, and the study of umbral algebra is called umbral calculus (see [1–3]).
The order of the nonzero power series is the smallest integer k for which the coefficient of does not vanish. If , then is called a delta series. If , then is called an invertible series (see ).
Let be polynomials in the variable x with degree n, and let and . Then there exists a unique sequence such that , where . The sequence is called the Sheffer sequence for , which is denoted by (see ).
with the usual convention about replacing by . In the special case, , are called the Euler numbers of order r.
with the usual convention about replacing by . In the special case, , are called the Bernoulli numbers of order r.
Recently, several researchers have studied the umbral calculus related to special polynomials. In this paper, we derive some interesting identities related to Bernoulli, Euler, and Abel polynomials arising from umbral calculus.
where is the Stirling number of the second kind. Therefore, by (2.2), we obtain the following theorem.
Therefore, by (2.4) and (2.7), we obtain the following theorem.
Therefore, by (2.9) and (2.10), we obtain the following lemma.
Therefore, by (2.12), we obtain the following proposition.
Therefore, by (2.14), we obtain the following theorem.
Therefore, by (2.24) and (2.25), we obtain the following theorem.
The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2013.
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