- Research Article
- Open Access
© D. S. Kim and K. H. Park. 2010
- Received: 16 February 2010
- Accepted: 13 April 2010
- Published: 20 May 2010
We derive eight basic identities of symmetry in three variables related to Euler polynomials and alternating power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundances of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identities are based on the -adic integral expression of the generating function for the Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating power sums.
- Positive Integer
- Continuous Function
- Generate Function
- Rational Number
- Nonnegative Integer
Many authors have done much work on identities of symmetry involving Bernoulli polynomials or Euler polynomials or -Bernoulli polynomials or -Euler polynomials. We let the reader refer to the papers in [1–20]. In connection with Bernoulli polynomials and power sums, these results were generalized in  to obtain identities of symmetry involving three variables in contrast to the previous works involving just two variables.
In this paper, we will produce 8 basic identities of symmetry in three variables , , related to Euler polynomials and alternating power sums (cf., (4.8), (4.9), (4.12), (4.16), (4.20), (4.23), (4.25), and (4.26)). These and most of their corollaries seem to be new, since there have been results only about identities of symmetry in two variables in the literature. These abundances of symmetries shed new light even on the existing identities. For instance, it has been known that (1.9) and (1.10) are equal and (1.11) and (1.12) are so (cf., [3, Theorems ]). In fact, (1.9)–(1.12) are all equal, as they can be derived from one and the same -adic integral. Perhaps, this was neglected to mention in . Also, we have a bunch of new identities in (1.13)–(1.16). All of these were obtained as corollaries (cf., Corollary 4.9, 4.12, 4.15) to some of the basic identities by specializing the variable as 1. Those would not be unearthed if more symmetries had not been available.
The derivations of identities will be based on the -adic integral expression of the generating function for the Euler polynomials in (1.5) and the quotient of integrals in (1.8) that can be expressed as the exponential generating function for the alternating power sums. We indebted this idea to the paper in .
Here we will introduce several types of quotients of -adic fermionic integrals on or from which some interesting identities follow owing to the built-in symmetries in , , . In the following, , , are all positive integers and all of the explicit expressions of integrals in (2.2), (2.4), (2.6), and (2.8) are obtained from the identity in (1.4).
All of the above -adic integrals of various types are invariant under all permutations of as one can see either from -adic integral representations in (2.1), (2.3), (2.5), and (2.7) or from their explicit evaluations in (2.2), (2.4), (2.6), and (2.8).
For Type , we may consider the analogous things to the ones in (a-0), (a-1), (a-2), and (a-3). However, these do not lead us to new identities. Indeed, if we substitute , , , respectively, for in (2.1), this amounts to replacing by in (2.3). So, upon replacing , , , respectively, by , , , and then dividing by , in each of the expressions of Theorem 4.1 through Corollary 4.15, we will get the corresponding symmetric identities for Type .
As we noted earlier in the last paragraph of Section 2, the various types of quotients of -adic fermionic integrals are invariant under any permutation of , , . So the corresponding expressions in Section 3 are also invariant under any permutation of , , . Thus our results about identities of symmetry will be immediate consequences of this observation.
However, not all permutations of an expression in Section 3 yield distinct ones. In fact, as these expressions are obtained by permuting , , in a single one labelled by them, they can be viewed as a group in a natural manner and hence it is isomorphic to a quotient of . In particular, the numbers of possible distinct expressions are , , or . (a-0), (a-1(1)), (a-1(2)), and (a-2(2)) give the full six identities of symmetry, (a-2(1)) and (a-2(3)) yield three identities of symmetry, and (c-0) and (c-1) give two identities of symmetry, while the expression in (a-3) yields no identities of symmetry.
However, (4.4) and (4.5) are equal to (4.26), as we can see by applying the permutations , , and for (4.4) and , , and for (4.5). Similarly, we see that (4.6) and (4.7) are equal to (4.27), by applying permutations , , and for (4.6) and , , and for (4.7).
This work was supported by National Foundation of Korea Grant funded by the Korean Government (2009-0072514).
- Deeba EY, Rodriguez DM: Stirling's series and Bernoulli numbers. The American Mathematical Monthly 1991, 98(5):423–426. 10.2307/2323860MathSciNetView ArticleMATHGoogle Scholar
- Howard FT: Applications of a recurrence for the Bernoulli numbers. Journal of Number Theory 1995, 52(1):157–172. 10.1006/jnth.1995.1062MathSciNetView ArticleMATHGoogle Scholar
- Kim T: Symmetry -adic invariant integral on for Bernoulli and Euler polynomials. Journal of Difference Equations and Applications 2008, 14(12):1267–1277. 10.1080/10236190801943220MathSciNetView ArticleMATHGoogle Scholar
- Kim T: Symmetry identities for the twisted generalized Euler polynomials. Advanced Studies in Contemporary Mathematics 2009, 19(2):151–155.MathSciNetMATHGoogle Scholar
- Kim T, Park KH, Hwang K-W: On the identities of symmetry for the -Euler polynomials of higher order. Advances in Difference Equations 2009, 2009:-9.Google Scholar
- Kim T: Symmetry of power sum polynomials and multivariate fermionic -adic invariant integral on . Russian Journal of Mathematical Physics 2009, 16(1):93–96. 10.1134/S1061920809010063MathSciNetView ArticleMATHGoogle Scholar
- Kim T: On the symmetries of the -Bernoulli polynomials. Abstract and Applied Analysis 2008, 2008:-7.Google Scholar
- Kim T: -Volkenborn integration. Russian Journal of Mathematical Physics 2002, 9(3):288–299.MathSciNetMATHGoogle Scholar
- Kim T: Note on the Euler -zeta functions. Journal of Number Theory 2009, 129(7):1798–1804. 10.1016/j.jnt.2008.10.007MathSciNetView ArticleMATHGoogle Scholar
- Kim T: On -adic interpolating function for -Euler numbers and its derivatives. Journal of Mathematical Analysis and Applications 2008, 339(1):598–608. 10.1016/j.jmaa.2007.07.027MathSciNetView ArticleMATHGoogle Scholar
- Kim T: On the -extension of Euler and Genocchi numbers. Journal of Mathematical Analysis and Applications 2007, 326(2):1458–1465. 10.1016/j.jmaa.2006.03.037MathSciNetView ArticleMATHGoogle Scholar
- Kim T: On the analogs of Euler numbers and polynomials associated with -adic -integral on at . Journal of Mathematical Analysis and Applications 2007, 331(2):779–792. 10.1016/j.jmaa.2006.09.027MathSciNetView ArticleMATHGoogle Scholar
- Kim T, Kim Y-H: On the symmetric properties for the generalized twisted Bernoulli polynomials. Journal of Inequalities and Applications 2009, 2009:-8.Google Scholar
- Kim T, Jang L-C, Kim Y-H, Hwang K-W: On the identities of symmetry for the generalized Bernoulli polynomials attached to of higher order. Journal of Inequalities and Applications 2009, 2009:-7.Google Scholar
- Ozden H, Cangul IN, Simsek Y: Multivariate interpolation functions of higher-order -Euler numbers and their applications. Abstract and Applied Analysis 2008, 2008:-16.Google Scholar
- Ozden H, Cangul IN, Simsek Y: Remarks on sum of products of -twisted Euler polynomials and numbers. Journal of Inequalities and Applications 2008, 2008:-8.Google Scholar
- Ozden H, Simsek Y: A new extension of -Euler numbers and polynomials related to their interpolation functions. Applied Mathematics Letters 2008, 21(9):934–939. 10.1016/j.aml.2007.10.005MathSciNetView ArticleGoogle Scholar
- Simsek Y: Complete sum of products of -extension of Euler polynomials and numbers. Journal of Difference Equations and Applications 2009.Google Scholar
- Tuenter HJH: A symmetry of power sum polynomials and Bernoulli numbers. The American Mathematical Monthly 2001, 108(3):258–261. 10.2307/2695389MathSciNetView ArticleMATHGoogle Scholar
- Yang S-L: An identity of symmetry for the Bernoulli polynomials. Discrete Mathematics 2008, 308(4):550–554. 10.1016/j.disc.2007.03.030MathSciNetView ArticleMATHGoogle Scholar
- Kim DS, Park KH: Identities of symmetry for Bernoulli polynomials arising from quotients of Volkenborn integrals invariant under S3, submittedGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.