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.
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.
- Sitaramachandrarao R, Davis B: Some identities involving the Riemann zeta function II. Indian J. Pure Appl. Math. 1986, 17: 1175–1186.MathSciNetGoogle Scholar
- 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 861797Google Scholar
- Dilcher K: Sums of products of Bernoulli numbers. J. Number Theory 1996, 60: 23–41. 10.1006/jnth.1996.0110MathSciNetView ArticleGoogle Scholar
- 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 269847Google Scholar
- Kim DS, Kim T, Choi J, Kim YH: Some identities on Bernoulli and Euler numbers. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 486158Google Scholar
- Kim DS, Kim T, Choi J, Kim YH: Identities involving q -Bernoulli and q -Euler numbers. Abstr. Appl. Anal. 2012., 2012: Article ID 674210Google Scholar
- 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 847901Google Scholar
- 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 856132Google Scholar
- Kim H-M, Kim DS: Arithmetic identities involving Bernoulli and Euler numbers. Int. J. Math. Math. Sci. 2012., 2012: Article ID 689797Google Scholar
- Kim M-S: On Euler numbers, polynomials and related p -adic integrals. J. Number Theory 2009, 129: 2166–2179. 10.1016/j.jnt.2008.11.004MathSciNetView ArticleGoogle Scholar
- Kim M-S: A note on sums of products of Bernoulli numbers. Appl. Math. Lett. 2011, 24(1):55–61. 10.1016/j.aml.2010.08.014MathSciNetView ArticleGoogle Scholar
- Kim T: On the analogs of Euler numbers and polynomials associated with p -adic q -integral on at . J. Math. Anal. Appl. 2007, 331: 779–792. 10.1016/j.jmaa.2006.09.027MathSciNetView ArticleGoogle Scholar
- Kim T, Kim DS, Bayad A, Rim S-H: Identities on the Bernoulli and the Euler numbers and polynomials. Ars Comb. 2012, CVII: 455–463.MathSciNetGoogle Scholar
- Lee I, Kim DS: Derivation of identities involving Bernoulli and Euler numbers. Int. J. Math. Math. Sci. 2012., 2012: Article ID 598543Google Scholar
- Petojević A: New sums of products of Bernoulli numbers. Integral Transforms Spec. Funct. 2008, 19: 105–114. 10.1080/10652460701541795MathSciNetView ArticleGoogle Scholar
- Petojević A, Srivastava HM: Computation of Euler’s type sums of the products of Bernoulli numbers. Appl. Math. Lett. 2009, 22: 796–801. 10.1016/j.aml.2008.06.040MathSciNetView ArticleGoogle Scholar
- Raabe JL: Zurückführung einiger Summen und bestmmtiem Integrale auf die Jacob-Bernoullische Function. J. Reine Angew. Math. 1851, 42: 348–367.MathSciNetView ArticleGoogle Scholar
- Simsek Y: q -analogue of twisted l -series and q -twisted Euler numbers. J. Number Theory 2005, 110: 267–278. 10.1016/j.jnt.2004.07.003MathSciNetView ArticleGoogle Scholar
- Srivastava HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 2000, 129: 77–84. 10.1017/S0305004100004412View ArticleGoogle Scholar
- Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht; 2001.View ArticleGoogle Scholar
- Srivastava HM, Pinter A: Remarks on some relationships between the Bernoulli and Euler polynomials. Appl. Math. Lett. 2004, 17: 375–380. 10.1016/S0893-9659(04)90077-8MathSciNetView ArticleGoogle Scholar
- Kim M-S, Hu S: Sums of products of Apostol-Bernoulli numbers. Ramanujan J. 2012, 28: 113–123. 10.1007/s11139-011-9340-zMathSciNetView ArticleGoogle Scholar
- Miki H: A relation between Bernoulli numbers. J. Number Theory 1978, 10: 297–302. 10.1016/0022-314X(78)90026-4MathSciNetView ArticleGoogle Scholar
- Matiyasevich, Y: Identities with Bernoulli numbers. http://logic.pdmi.ras.ru/~yumat/Journal/Bernoulli/bernulli.htmGoogle Scholar
- Dunne, GV, Schubert, C: Bernoulli number identities from quantum field theory. Preprint (2004). arXiv:math.NT/0406610Google Scholar
- Crabb MC: The Miki-Gessel Bernoulli number identity. Glasg. Math. J. 2005, 47: 327–328. 10.1017/S0017089505002545MathSciNetView ArticleGoogle Scholar
- Gessel IM: On Miki’s identity for Bernoulli numbers. J. Number Theory 2005, 110: 75–82. 10.1016/j.jnt.2003.08.010MathSciNetView ArticleGoogle Scholar
- Sun Z-W, Pan H: Identities concerning Bernoulli and Euler polynomials. Acta Arith. 2006, 125: 21–39. 10.4064/aa125-1-3MathSciNetView ArticleGoogle Scholar
- Cohen H Graduate Texts in Mathematics 240. In Number Theory, Vol. II: Analytic and Modern Tools. Springer, New York; 2007.Google Scholar
- Robert AM Graduate Texts in Mathematics 198. In A Course in p-Adic Analysis. Springer, New York; 2000.View ArticleGoogle Scholar
- 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. 1985, 39: 113–125.MathSciNetGoogle Scholar
- Osipov JV: p -adic zeta functions. Usp. Mat. Nauk 1979, 34: 209–210. (in Russian)Google Scholar
- Maïga H: Some identities and congruences concerning Euler numbers and polynomials. J. Number Theory 2010, 130: 1590–1601. 10.1016/j.jnt.2010.01.019MathSciNetView ArticleGoogle Scholar
- Kim M-S, Hu S: On p -adic Hurwitz-type Euler zeta functions. J. Number Theory 2012, 132: 2977–3015. 10.1016/j.jnt.2012.05.037MathSciNetView ArticleGoogle Scholar
- 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.pdfGoogle Scholar
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