- Research Article
- Open Access
Identities of Symmetry for Euler Polynomials Arising from Quotients of Fermionic Integrals Invariant under
© 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
In what follows, we will always assume that the -adic fermionic integrals of the various exponential functions on are defined for (cf., (1.3)), and therefore it will not be mentioned.
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).
In the following are all odd positive integers except for (a-0) and (c-0), where they are any positive integers.
(a-1)Here we write in two different ways:
(a-2)Here we write in three different ways:
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.
But, by interchanging and , we see that (4.1), (4.2), and (4.3) are, respectively, equal to (4.15), (4.16), and (4.17).
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).
Putting in (4.9), we get the following corollary.
Letting further in (4.10), we have the following corollary.
Letting in (4.12), we obtain alternative expressions for the identities in (4.10).
Putting further in (4.13), we have the alternative expressions for the identities for (4.11).
Putting in (4.15)–(4.17), we get the following corollary.
Putting in (4.20), we obtain the following corollary. In Section 1, the identities in (4.18), (4.21), and (4.24) are combined to give those in (1.9)–(1.16).
Letting further in (4.21), we get the following corollary. This is the multiplication formula for Euler polynomials together with the relatively new identity mentioned in (4.19).
Letting in (4.23), we have the following corollary.
Putting in (4.26) and (4.27), we get the following corollary.
This work was supported by National Foundation of Korea Grant funded by the Korean Government (2009-0072514).
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