A note on higher-order Bernoulli polynomials
© Kim and Kim; licensee Springer. 2013
Received: 23 September 2012
Accepted: 28 February 2013
Published: 19 March 2013
Let be the -dimensional vector space over Q. From the property of the basis for the space , we derive some interesting identities of higher-order Bernoulli polynomials.
with the usual convention about replacing by . In the special case, , are called the n th Bernoulli numbers of order r.
By (2) and (3), we see that is a monic polynomial of degree n with coefficients in Q.
Then we see that (i) , (ii) , (iii) .
Let be the -dimensional vector space over Q. Probably, is the most natural basis for this space. But is also a good basis for the space for our purpose of arithmetical and combinatorial applications.
In this paper, we develop methods for uniquely determining from the information of . From those methods, we derive some interesting identities of higher-order Bernoulli polynomials.
2 Higher-order Bernoulli polynomials
For , by (1), we get (). Let .
Case 1. Let . Then for all .
- (i)For , by (15) we get(17)
- (ii)For , by (15) we see that(18)
Therefore, by (7), (16), (17) and (18), we obtain the following theorem.
- (a)For , we have
- (b)For , we have
Therefore, by Theorem 1 and (19), we obtain the following corollary.
Thus, by Theorem 1 and (20), we obtain the following corollary.
Thus, by Theorem 1 and (22), we obtain the following theorem.
Therefore, by (24), we obtain the following corollary.
- (a)For with , we have
- (b)In particular, , we get
Therefore, by Theorem 1, (21) and (25), we obtain the following theorem.
Let () be Euler polynomials of order s. Then can be expressed as a linear combination of .
Assume that with .
Therefore, by Theorem 1 and (26), we obtain the following theorem.
By Theorem 1 and (27), we obtain the following theorem.
3 Further remarks
- (i)For , we have(40)
- (ii)For , we have(41)
Thus, by (38), (39), (40) and (41), we can determine .
This paper is supported in part by the Research Grant of Kwangwoon University in 2013.
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