Abstract
In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulas for calculating the poly-Bernoulli numbers of the second kind and the Stirling numbers of the second kind.
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1 Introduction
As is well known, the Bernoulli polynomials of the second kind are defined by the generating function to be
When , are called the Bernoulli numbers of the second kind. The first few Bernoulli numbers of the second kind are , , , , , .
From (1), we have
where (). The Stirling number of the second kind is defined by
The ordinary Bernoulli polynomials are given by
When , are called Bernoulli numbers.
It is well known that the classical poly-logarithmic function is given by
For , . The Stirling number of the first kind is defined by
In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulas for calculating the poly-Bernoulli number and polynomial and the Stirling number of the second kind.
2 Poly-Bernoulli numbers and polynomials of the second kind
For , we consider the poly-Bernoulli polynomials of the second kind:
When , are called the poly-Bernoulli numbers of the second kind.
Indeed, for , we have
By (7) and (8), we get
It is well known that
where are the Bernoulli polynomials of order α which are given by the generating function to be
By (1) and (10), we get
Now, we observe that
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Thus, by (11), we get
Therefore, by (12), we obtain the following theorem.
Theorem 2.1 For we have
From (11), we have
We observe that
Thus, by (10) and (14), we get
Therefore, by (15), we obtain the following theorem.
Theorem 2.2 For , we have
By (7), we get
Therefore, by (16), we obtain the following theorem.
Theorem 2.3 For , we have
From (13), we have
Therefore, by (17), we obtain the following theorem.
Theorem 2.4 For , we have
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The present research has been conducted by the Research Grant of Kwangwoon University in 2014.
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Kim, T., Kwon, H.I., Lee, S.H. et al. A note on poly-Bernoulli numbers and polynomials of the second kind. Adv Differ Equ 2014, 219 (2014). https://doi.org/10.1186/1687-1847-2014-219
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DOI: https://doi.org/10.1186/1687-1847-2014-219