- Open Access
Apostol-Euler polynomials arising from umbral calculus
© Kim et al.; licensee Springer. 2013
- Received: 3 April 2013
- Accepted: 10 October 2013
- Published: 8 November 2013
In this paper, by using the orthogonality type as defined in the umbral calculus, we derive an explicit formula for several well-known polynomials as a linear combination of the Apostol-Euler polynomials.
- Bernoulli polynomial
- Bessel polynomial
- Euler polynomial
- Frobenius-Euler polynomial
- umbral calculus
where is the Kronecker symbol. Let with . From (1.3), we have . So, the map is a vector space isomorphic from onto ℋ. Henceforth, ℋ is thought of as a set of both formal power series and linear functionals. We call ℋ umbral algebra. The umbral calculus is the study of umbral algebra.
Let . The smallest integer k for which the coefficient of does not vanish is called the order of and is denoted by (see [1–4]). If , , then is called a delta, an invertible series, respectively. For given two power series such that and , there exists a unique sequence of polynomials with (this condition sometimes is called orthogonality type) for all . The sequence is called the Sheffer sequence for which is denoted by (see [1–4]).
In the next section, we present our main theorem and its applications. More precisely, by using the orthogonality type, we write any polynomial in as a linear combination of the Apostol-Euler polynomials. Several applications related to Bernoulli, Euler and Frobenius-Euler polynomials are derived.
Hence, we can state the following result.
which implies the following identity.
which, by (1.15), implies the following identity.
which, by (1.18), implies the following identity.
which, by (1.21), implies the following identity.
which implies the following identity.
We end by noting that if we substitute in any of our corollaries, then we get the well-known value of the polynomial . For instance, by setting , the last corollary gives that , as expected.
This paper is supported in part by the Research Grant of Kwangwoon University in 2013.
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