Abstract
Using padic integral, many new convolution identities involving Bernoulli, Euler and Genocchi numbers are given.
MSC: 11B68, 11S80.
Keywords:
Bernoulli numbers; Euler numbers; sums of products; fermionic padic integral; Volkenborn integral1 Introduction
The study of the identities involving Bernoulli numbers and polynomials, Euler numbers and polynomials has a long history. More than 250 years ago, Euler discovered the following sums of products identity involving Bernoulli numbers:
(see [1]).
This identity has been extended by many authors in different directions (see [221]). Recently, Kim and Hu [22] obtained an analogy of Euler’s sums of products identity for ApostolBernoulli numbers.
In 1978, Miki [23] obtained the following wellknown identity, which is now known as Miki’s identity:
In 1997, Matiyasevich [24] found another identity of this type
In 2004, Dunne and Schubert [25] 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 [26] showed that Gessel’s generalization of Miki’s identity [27] is a direct consequence of a functional equation for the generating function. During the same year, Sun and Pan [28] 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 [[29], Chapter 11], the padic 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.
For example, in Exercises 2 and 3 of [[29], Chapter 11], Cohen demonstrated a method to prove the following reciprocity formula for Bernoulli numbers using padic integral:
(see Exercise 3(c) of [[29], Chapter 11]).
Recently, using padic integral, Kim et al.[8] proved several identities on Bernoulli and Euler numbers. More comprehensive coverage can be found in the monographs by Chio et al.[2], Kim et al.[4], Kim et al.[5], Kim et al.[6], Kim et al.[7], Kim and Kim [9], Kim et al.[13], Lee and Kim [14].
In this paper, following the methods of [8], we shall further provide many new convolution identities involving Bernoulli, Euler and Genocchi numbers.
Let , and be the nth 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 .
In fact, the following identities are shown in this paper:
This paper is organized as follows. In the next section, we recall the fundamental results between padic 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 higherorder 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 padic analysis
In this section, we recall the fundamental results between padic integral and Bernoulli and Euler numbers, and we see that the properties of padic integrals may imply most wellknown 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 padic integers, the field of padic numbers and the field of padic completion of the algebraic closure of , respectively. ℕ denotes the set of natural numbers and denotes .
Let be the space of uniformly (or strictly) differentiable function on . Then the Volkenborn integral of f is defined by
and this limit always exists when (see [[30], p.264]). For such functions we have
where and . By (2.2), we have the following Volkenborn padic integral representation of the generating function of Bernoulli polynomials :
which is equivalent to
(see [12]).
Setting in (2.4), we obtain the Volkenborn padic integral representation of the nth Bernoulli numbers
Thus from the binomial theorem and (2.4), we get
where
The recurrence formula of is given by
with the usual convention of denoting by .
By (2.6) and (2.8), we have
The fermionic padic integral on is defined by
where . This integral was introduced by Kim [12] in order to derive useful formulas involving the Euler numbers and polynomials. It has also been defined independently by Shiratani and Yamamoto [31] in order to interpolate the Euler numbers padically. Osipov [32] gave a new proof of the existence of the KubotaLeopoldt padic zeta function by using the integral representation
where , , , , and . Note that when , is the fermionic padic integral on . Recently, the fermionic padic 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 [10]; it has also been used by Maïga [33] to give some new identities and congruences concerning Euler numbers and polynomials.
From the definition of , we have
where (see [[12], Lemma 1]). Thus from induction, we have the following fermionic padic integral equation:
(see [[12], Theorem 2]).
Using formula (2.12), we have the following fermionic padic integral representation of the generating function of Euler polynomials :
namely
(see [[34], Proposition 2.1]).
Setting in (2.16), we obtain the fermionic padic integral representation of the nth Euler numbers
(comparing with (2.5)).
Therefore, by the binomial theorem, (2.16), and (2.17), we have
The recurrence formula of is given by
with the usual convention of denoting by .
By (2.18) and (2.19), we have
The padic integral also implies the following reflection symmetric relations for the Bernoulli and the Euler polynomials (see formulas (2.12) and (2.13) in [34]):
3 Identities involving Bernoulli and Euler numbers
In this section, we prove new identities involving Bernoulli and Euler numbers.
Proof Putting in (2.12), by (2.17), we have
By (2.22), with , and (2.20), we obtain
substituting it to (3.1), we have the desired result. □
The following wellknown fact on Euler polynomials may also be established using padic integral. We refer to Corollary 1.1 in [35] for another proof.
Proof We prove our result following the method of the proof for Theorem 2.1 in [8].
By (2.17) and (2.18), we have
On the other hand, using (2.18), (2.22) and Lemma 3.1, we obtain
where is the Kronecker symbol.
Comparing (3.2) and (3.3), then using (2.18), with , and (2.20), we have
Putting on the lefthand side of (3.4), we have , hence . Replacing n by 2n (), the righthand side becomes , which completes the proof. □
Proof The proof goes the same way as in the above theorem.
By (2.5) and (2.6), we have
On the other hand, using (2.6), (2.21) and Lemma 3.1, we obtain
Comparing (3.5) and (3.6), then using (2.6), with , and (2.9), we have
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 [8].
Putting in (2.6), (2.18), and using (2.9), (2.20), we get
By the linear property of padic integral and (2.17), we have the following fermionic padic integral representation of the product of Bernoulli and Euler polynomials:
Further, by (2.21), (2.22), Lemma 3.1 and (3.13), we have
Comparing (3.14) and (3.15), then using (3.13), we have
Replacing m by and n by in (3.16), respectively, we have
and
Finally, replacing n by in (3.17) and m by 2m in (3.18), using Theorem 3.2, we get our result. □
Remark 3.6 Putting in (3.17), we recover Theorem 3.3. Setting in (3.18), we obtain
This identity is trivial, because by Theorem 3.2, the th terms are identical to zero.
4 Identities involving higherorder Bernoulli and Euler numbers
In this section, we prove new identities involving higherorder Bernoulli and Euler numbers and polynomials.
The higherorder Bernoulli polynomials and the higherorder Euler polynomials are defined by the following generating functions:
and
respectively. For , we have and , where and denote the higherorder Bernoulli and Euler numbers, respectively (see [[20], Section 1.6] and [[21], (16), (17)]). Moreover, from (4.1) and (4.2), we have
Proof Letting () in (2.14), we have .
By (2.13), we have
By (2.20), (2.22) with , we have , and we obtain the assertion of the lemma. □
Proof From (2.17), (4.3) and the linear property of padic integral, we obtain the following fermionic padic integral representation of the higherorder Bernoulli polynomials:
On the other hand, by (4.4) and Lemma 4.1, we have
Clearly,
Comparing (4.8) and (4.9), then using (4.10) and (4.3) with , we obtain
Since , therefore, by (4.11), we have
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. □
Proof First we have the following fermionic padic integral representation of the product of higherorder Bernoulli and Euler polynomials:
On the other hand, by (4.4), (4.6), Lemma 4.1, by (4.3) and (4.5) with , we have
Comparing (4.12) and (4.13), and also noticing that and , we have
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 higherorder Genocchi polynomials are defined by the generating function
The Genocchi polynomials are given by . For , we have the Genocchi numbers , i.e., . Letting , we also have , where denotes the higherorder 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.
From the generating function (5.1), it is easy to deduce that
Therefore, by the fermionic padic integral representation of , (5.2) and (5.3), we have
For , by (5.4), we have . From (5.4), it is easily seen that
Also, by the fermionic padic representations of and , we have
which is equivalent to
Competing interests
The authors declare that they have no competing interest.
Authors’ contributions
All authors contributed equally in this paper. They read and approved the final manuscript.
Acknowledgements
This work was supported by the Kyungnam University Foundation Grant, 2013.
References

Sitaramachandrarao, R, Davis, B: Some identities involving the Riemann zeta function II. Indian J. Pure Appl. Math.. 17, 1175–1186 (1986)

Choi, J, Kim, DS, Kim, T, Kim, YH: A note on some identities of FrobeniusEuler numbers and polynomials. Int. J. Math. Math. Sci.. 2012, (2012) Article ID 861797

Dilcher, K: Sums of products of Bernoulli numbers. J. Number Theory. 60, 23–41 (1996). Publisher Full Text

Kim, DS, Dolgy, DV, Kim, HM, Lee, SH, Kim, T: Integral formulae of Bernoulli polynomials. Discrete Dyn. Nat. Soc.. 2012, (2012) Article ID 269847

Kim, DS, Kim, T, Choi, J, Kim, YH: Some identities on Bernoulli and Euler numbers. Discrete Dyn. Nat. Soc.. 2012, (2012) Article ID 486158

Kim, DS, Kim, T, Choi, J, Kim, YH: Identities involving qBernoulli and qEuler numbers. Abstr. Appl. Anal.. 2012, (2012) Article ID 674210

Kim, DS, Kim, T, Dolgy, DV, Lee, SH, Rim, SH: Some properties and identities of Bernoulli and Euler polynomials associated with padic integral on . Abstr. Appl. Anal.. 2012, (2012) Article ID 847901

Kim, DS, Kim, T, Lee, SH, Dolgy, DV, Rim, SH: Some new identities on the Bernoulli and Euler numbers. Discrete Dyn. Nat. Soc.. 2011, (2011) Article ID 856132

Kim, HM, Kim, DS: Arithmetic identities involving Bernoulli and Euler numbers. Int. J. Math. Math. Sci.. 2012, (2012) Article ID 689797

Kim, MS: On Euler numbers, polynomials and related padic integrals. J. Number Theory. 129, 2166–2179 (2009). Publisher Full Text

Kim, MS: A note on sums of products of Bernoulli numbers. Appl. Math. Lett.. 24(1), 55–61 (2011). Publisher Full Text

Kim, T: On the analogs of Euler numbers and polynomials associated with padic qintegral on at . J. Math. Anal. Appl.. 331, 779–792 (2007). Publisher Full Text

Kim, T, Kim, DS, Bayad, A, Rim, SH: Identities on the Bernoulli and the Euler numbers and polynomials. Ars Comb.. CVII, 455–463 (2012)

Lee, I, Kim, DS: Derivation of identities involving Bernoulli and Euler numbers. Int. J. Math. Math. Sci.. 2012, (2012) Article ID 598543

Petojević, A: New sums of products of Bernoulli numbers. Integral Transforms Spec. Funct.. 19, 105–114 (2008). Publisher Full Text

Petojević, A, Srivastava, HM: Computation of Euler’s type sums of the products of Bernoulli numbers. Appl. Math. Lett.. 22, 796–801 (2009). Publisher Full Text

Raabe, JL: Zurückführung einiger Summen und bestmmtiem Integrale auf die JacobBernoullische Function. J. Reine Angew. Math.. 42, 348–367 (1851)

Simsek, Y: qanalogue of twisted lseries and qtwisted Euler numbers. J. Number Theory. 110, 267–278 (2005). Publisher Full Text

Srivastava, HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc.. 129, 77–84 (2000). Publisher Full Text

Srivastava, HM, Choi, J: Series Associated with the Zeta and Related Functions, Kluwer Academic, Dordrecht (2001)

Srivastava, HM, Pinter, A: Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett.. 17, 375–380 (2004). Publisher Full Text

Kim, MS, Hu, S: Sums of products of ApostolBernoulli numbers. Ramanujan J.. 28, 113–123 (2012). Publisher Full Text

Miki, H: A relation between Bernoulli numbers. J. Number Theory. 10, 297–302 (1978). Publisher Full Text

Matiyasevich, Y: Identities with Bernoulli numbers. http://logic.pdmi.ras.ru/~yumat/Journal/Bernoulli/bernulli.htm

Dunne, GV, Schubert, C: Bernoulli number identities from quantum field theory. Preprint (2004). arXiv:math.NT/0406610

Crabb, MC: The MikiGessel Bernoulli number identity. Glasg. Math. J.. 47, 327–328 (2005). Publisher Full Text

Gessel, IM: On Miki’s identity for Bernoulli numbers. J. Number Theory. 110, 75–82 (2005). Publisher Full Text

Sun, ZW, Pan, H: Identities concerning Bernoulli and Euler polynomials. Acta Arith.. 125, 21–39 (2006). Publisher Full Text

Cohen, H: Number Theory, Vol. II: Analytic and Modern Tools, Springer, New York (2007)

Robert, AM: A Course in pAdic Analysis, Springer, New York (2000)

Shiratani, K, Yamamoto, S: On a padic interpolation function for the Euler numbers and its derivatives. Mem. Fac. Sci., Kyushu Univ., Ser. A, Math.. 39, 113–125 (1985)

Osipov, JV: padic zeta functions. Usp. Mat. Nauk. 34, 209–210 (1979) (in Russian)

Maïga, H: Some identities and congruences concerning Euler numbers and polynomials. J. Number Theory. 130, 1590–1601 (2010). Publisher Full Text

Kim, MS, Hu, S: On padic Hurwitztype Euler zeta functions. J. Number Theory. 132, 2977–3015 (2012). Publisher Full Text

Sun, ZW: Introduction to Bernoulli and Euler polynomials. A Lecture Given in Taiwan on June 6, 2002. http://math.nju.edu.cn/~zwsun/BerE.pdf