New identities involving Bernoulli, Euler and Genocchi numbers
© Hu et al.; licensee Springer. 2013
Received: 26 October 2012
Accepted: 5 March 2013
Published: 26 March 2013
Using p-adic integral, many new convolution identities involving Bernoulli, Euler and Genocchi numbers are given.
KeywordsBernoulli numbers Euler numbers sums of products fermionic p-adic integral Volkenborn integral
for every , where .
for any .
In 2004, Dunne and Schubert  obtained the convolution identities for sums of products of Bernoulli numbers motivated by the role of these identities in quantum field theory and string theory. In 2006, Crabb  showed that Gessel’s generalization of Miki’s identity  is a direct consequence of a functional equation for the generating function. During the same year, Sun and Pan  established two general identities involving Bernoulli and Euler polynomials, which imply both Miki and Matiyasevich’s identities.
According to statement by Cohen in the first paragraph of [, Chapter 11], the p-adic functions with nice properties are powerful tools for studying many results of classical number theory in a straightforward manner, for instance, strengthening of almost all the arithmetic results on Bernoulli numbers.
(see Exercise 3(c) of [, Chapter 11]).
Recently, using p-adic integral, Kim et al.  proved several identities on Bernoulli and Euler numbers. More comprehensive coverage can be found in the monographs by Chio et al. , Kim et al. , Kim et al. , Kim et al. , Kim et al. , Kim and Kim , Kim et al. , Lee and Kim .
In this paper, following the methods of , we shall further provide many new convolution identities involving Bernoulli, Euler and Genocchi numbers.
Let , and be the n th Bernoulli, Euler and Genocchi polynomials, respectively. In what follows, we use to denote the special value of at 0, that is, .
Let be the Kronecker symbol defined by and for .
This paper is organized as follows. In the next section, we recall the fundamental results between p-adic integral and Bernoulli and Euler numbers. Then using these results we prove new identities (identities (1.2), (1.3), (1.4), (1.5), (1.7), (1.8), (1.9)) involving Bernoulli and Euler numbers in Section 3, prove new identities (identities (1.10), (1.11), (1.12)) involving higher-order Bernoulli and Euler numbers and polynomials in Section 4, and also prove a new identity (identity (1.13)) involving both Bernoulli, Euler and Genocchi numbers in the final section.
2 p-adic analysis
In this section, we recall the fundamental results between p-adic integral and Bernoulli and Euler numbers, and we see that the properties of p-adic integrals may imply most well-known facts on Bernoulli numbers and polynomials, Euler numbers and polynomials.
We assume p is an odd prime number. The symbols , and denote the rings of p-adic integers, the field of p-adic numbers and the field of p-adic completion of the algebraic closure of , respectively. ℕ denotes the set of natural numbers and denotes .
with the usual convention of denoting by .
where , , , , and . Note that when , is the fermionic p-adic integral on . Recently, the fermionic p-adic integral on has been used by the third author to give a brief proof of Stein’s classical result on Euler numbers modulo power of two ; it has also been used by Maïga  to give some new identities and congruences concerning Euler numbers and polynomials.
(see [, Theorem 2]).
(see [, Proposition 2.1]).
(comparing with (2.5)).
with the usual convention of denoting by .
3 Identities involving Bernoulli and Euler numbers
In this section, we prove new identities involving Bernoulli and Euler numbers.
substituting it to (3.1), we have the desired result. □
The following well-known fact on Euler polynomials may also be established using p-adic integral. We refer to Corollary 1.1 in  for another proof.
Theorem 3.2 for .
Proof We prove our result following the method of the proof for Theorem 2.1 in .
where is the Kronecker symbol.
Putting on the left-hand side of (3.4), we have , hence . Replacing n by 2n (), the right-hand side becomes , which completes the proof. □
Proof The proof goes the same way as in the above theorem.
Letting () in this identity, we get our result. □
Taking , in (3.7) and (3.8), respectively, we obtain the following corollary.
Proof of Theorem 3.4 We prove our result following the method of the proof for Theorem 2.3 in .
Finally, replacing n by in (3.17) and m by 2m in (3.18), using Theorem 3.2, we get our result. □
This identity is trivial, because by Theorem 3.2, the th terms are identical to zero.
4 Identities involving higher-order Bernoulli and Euler numbers
In this section, we prove new identities involving higher-order Bernoulli and Euler numbers and polynomials.
Proof Letting () in (2.14), we have .
By (2.20), (2.22) with , we have , and we obtain the assertion of the lemma. □
which is the required result. □
Proof This follows from the same process as in the proof of Theorem 4.2 by using (2.17), (4.5), (4.6) and Lemma 4.1. □
which is the assertion of the theorem. □
5 Further remarks and observations
In this section, we show that the same methods as in Sections 3 and 4 can be used to obtain a new identity involving both Bernoulli, Euler and Genocchi numbers.
The Genocchi polynomials are given by . For , we have the Genocchi numbers , i.e., . Letting , we also have , where denotes the higher-order Genocchi numbers.
The generating function of Genocchi polynomials is similar to those of Bernoulli and Euler polynomials, so it may be expected that the Genocchi numbers also satisfy similar identities as those established in Sections 3 and 4.
This work was supported by the Kyungnam University Foundation Grant, 2013.
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