Research

# New identities involving Bernoulli, Euler and Genocchi numbers

Su Hu1, Daeyeoul Kim2 and Min-Soo Kim3*

Author Affiliations

1 Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology (KAIST), 373-1 Guseong-dong, Yuseong-gu, Daejeon, 305-701, Republic of Korea

2 National Institute for Mathematical Sciences, Yuseong-daero 1689-gil, Yuseong-gu, Daejeon, 305-811, Republic of Korea

3 Division of Cultural Education, Kyungnam University, 7 (Woryeong-dong) kyungnamdaehak-ro, Masanhappo-gu, Changwon-si, Gyeongsangnam-do, 631-701, Republic of Korea

For all author emails, please log on.

Advances in Difference Equations 2013, 2013:74  doi:10.1186/1687-1847-2013-74

 Received: 26 October 2012 Accepted: 5 March 2013 Published: 26 March 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Using p-adic 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 p-adic integral; Volkenborn integral

### 1 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:

(1.1)

(see [1]).

This identity has been extended by many authors in different directions (see [2-21]). Recently, Kim and Hu [22] obtained an analogy of Euler’s sums of products identity for Apostol-Bernoulli numbers.

In 1978, Miki [23] obtained the following well-known identity, which is now known as Miki’s identity:

for every  , where .

In 1997, Matiyasevich [24] found another identity of this type

for any  .

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 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.

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 p-adic integral:

(see Exercise 3(c) of [[29], Chapter 11]).

Recently, using p-adic 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:

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

(1.7)

(1.8)

(1.9)

(1.10)

(1.11)

(1.12)

(1.13)

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.

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 .

Let be the space of uniformly (or strictly) differentiable function on . Then the Volkenborn integral of f is defined by

(2.1)

and this limit always exists when (see [[30], p.264]). For such functions we have

(2.2)

where and . By (2.2), we have the following Volkenborn p-adic integral representation of the generating function of Bernoulli polynomials :

(2.3)

which is equivalent to

(2.4)

(see [12]).

Setting in (2.4), we obtain the Volkenborn p-adic integral representation of the nth Bernoulli numbers

(2.5)

(see [12,30]).

Thus from the binomial theorem and (2.4), we get

(2.6)

where

(2.7)

The recurrence formula of is given by

(2.8)

with the usual convention of denoting by .

By (2.6) and (2.8), we have

(2.9)

The fermionic p-adic integral on is defined by

(2.10)

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 p-adically. Osipov [32] gave a new proof of the existence of the Kubota-Leopoldt p-adic zeta function by using the integral representation

(2.11)

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 [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

(2.12)

where (see [[12], Lemma 1]). Thus from induction, we have the following fermionic p-adic integral equation:

(2.13)

where and

(2.14)

(see [[12], Theorem 2]).

Using formula (2.12), we have the following fermionic p-adic integral representation of the generating function of Euler polynomials :

(2.15)

namely

(2.16)

(see [[34], Proposition 2.1]).

Setting in (2.16), we obtain the fermionic p-adic integral representation of the nth Euler numbers

(2.17)

(comparing with (2.5)).

Therefore, by the binomial theorem, (2.16), and (2.17), we have

(2.18)

The recurrence formula of is given by

(2.19)

with the usual convention of denoting by .

By (2.18) and (2.19), we have

(2.20)

The p-adic integral also implies the following reflection symmetric relations for the Bernoulli and the Euler polynomials (see formulas (2.12) and (2.13) in [34]):

(2.21)

(2.22)

### 3 Identities involving Bernoulli and Euler numbers

In this section, we prove new identities involving Bernoulli and Euler numbers.

Lemma 3.1For, we have

Proof Putting in (2.12), by (2.17), we have

(3.1)

By (2.22), with , and (2.20), we obtain

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 [35] for another proof.

Theorem 3.2for.

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

(3.2)

On the other hand, using (2.18), (2.22) and Lemma 3.1, we obtain

(3.3)

where is the Kronecker symbol.

Comparing (3.2) and (3.3), then using (2.18), with , and (2.20), we have

(3.4)

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. □

Theorem 3.3For, we have

Proof The proof goes the same way as in the above theorem.

By (2.5) and (2.6), we have

(3.5)

On the other hand, using (2.6), (2.21) and Lemma 3.1, we obtain

(3.6)

Comparing (3.5) and (3.6), then using (2.6), with , and (2.9), we have

Letting () in this identity, we get our result. □

Theorem 3.4For, we have

(3.7)

(3.8)

Taking , in (3.7) and (3.8), respectively, we obtain the following corollary.

Corollary 3.5For, we have

(3.9)

(3.10)

(3.11)

(3.12)

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

(3.13)

By the linear property of p-adic integral and (2.17), we have the following fermionic p-adic integral representation of the product of Bernoulli and Euler polynomials:

(3.14)

Further, by (2.21), (2.22), Lemma 3.1 and (3.13), we have

(3.15)

Comparing (3.14) and (3.15), then using (3.13), we have

(3.16)

Replacing m by and n by in (3.16), respectively, we have

(3.17)

and

(3.18)

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 higher-order Bernoulli and Euler numbers

In this section, we prove new identities involving higher-order Bernoulli and Euler numbers and polynomials.

The higher-order Bernoulli polynomials and the higher-order Euler polynomials are defined by the following generating functions:

(4.1)

and

(4.2)

respectively. For , we have and , where and denote the higher-order Bernoulli and Euler numbers, respectively (see [[20], Section 1.6] and [[21], (16), (17)]). Moreover, from (4.1) and (4.2), we have

(4.3)

(4.4)

(4.5)

(4.6)

Lemma 4.1Forand, we have

Proof Letting () in (2.14), we have .

By (2.13), we have

(4.7)

By (2.20), (2.22) with , we have , and we obtain the assertion of the lemma. □

Theorem 4.2Forand, we have

Proof From (2.17), (4.3) and the linear property of p-adic integral, we obtain the following fermionic p-adic integral representation of the higher-order Bernoulli polynomials:

(4.8)

On the other hand, by (4.4) and Lemma 4.1, we have

(4.9)

Clearly,

(4.10)

Comparing (4.8) and (4.9), then using (4.10) and (4.3) with , we obtain

(4.11)

Since , therefore, by (4.11), we have

which is the required result. □

Theorem 4.3Forand, we have

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. □

Theorem 4.4Forand, we have

Proof First we have the following fermionic p-adic integral representation of the product of higher-order Bernoulli and Euler polynomials:

(4.12)

On the other hand, by (4.4), (4.6), Lemma 4.1, by (4.3) and (4.5) with , we have

(4.13)

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 higher-order Genocchi polynomials are defined by the generating function

(5.1)

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.

From the generating function (5.1), it is easy to deduce that

(5.2)

(5.3)

Therefore, by the fermionic p-adic integral representation of , (5.2) and (5.3), we have

(5.4)

For , by (5.4), we have . From (5.4), it is easily seen that

(5.5)

Also, by the fermionic p-adic representations of and , we have

(5.6)

which is equivalent to

where .

### 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

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

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

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

4. Kim, DS, Dolgy, DV, Kim, H-M, Lee, S-H, Kim, T: Integral formulae of Bernoulli polynomials. Discrete Dyn. Nat. Soc.. 2012, (2012) Article ID 269847

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

6. Kim, DS, Kim, T, Choi, J, Kim, YH: Identities involving q-Bernoulli and q-Euler numbers. Abstr. Appl. Anal.. 2012, (2012) Article ID 674210

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

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

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

10. Kim, M-S: On Euler numbers, polynomials and related p-adic integrals. J. Number Theory. 129, 2166–2179 (2009). Publisher Full Text

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

12. Kim, T: On the analogs of Euler numbers and polynomials associated with p-adic q-integral on at . J. Math. Anal. Appl.. 331, 779–792 (2007). Publisher Full Text

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

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

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

16. 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

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

18. Simsek, Y: q-analogue of twisted l-series and q-twisted Euler numbers. J. Number Theory. 110, 267–278 (2005). Publisher Full Text

19. 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

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

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

22. Kim, M-S, Hu, S: Sums of products of Apostol-Bernoulli numbers. Ramanujan J.. 28, 113–123 (2012). Publisher Full Text

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

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

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

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

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

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

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

30. Robert, AM: A Course in p-Adic Analysis, Springer, New York (2000)

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

32. Osipov, JV: p-adic zeta functions. Usp. Mat. Nauk. 34, 209–210 (1979) (in Russian)

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

34. Kim, M-S, Hu, S: On p-adic Hurwitz-type Euler zeta functions. J. Number Theory. 132, 2977–3015 (2012). Publisher Full Text

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