Identities of symmetry for generalized Euler polynomials

In this paper, we derive eight basic identities of symmetry in three variables related to generalized Euler polynomials and alternating generalized power sums. All of these are new, since there have been results only about identities of symmetry in two variables. The derivations of identities are based on the $p$-adic fermionic integral expression of the generating function for the generalized Euler polynomials and the quotient of integrals that can be expressed as the exponential generating function for the alternating generalized power sums.


Introduction and preliminaries
Let p be a fixed odd prime. Throughout this paper, Z p , Q p , C p will respectively denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of the algebraic closure of Q p . Let d be a fixed odd positive integer. Then we let X = X d = lim ← − N Z/dp N Z, and let π : X → Z p be the map given by the inverse limit of the natural maps Z/dp N Z → Z/p N Z.
If g is a function on Z p , we will use the same notation to denote the function g • π.
Let χ : (Z/dZ) * → Q * be a (primitive) Dirichlet character of conductor d. Then it will be pulled back to X via the natural map X → Z/dZ. Here we fix, once and for all, an imbedding Q → C p , so that χ is regarded as a map of X to C p (cf. [8]).
For a continuous function f : X → C p , the p-adic fermionic integral of f is defined by Then it is easy to see that and that, for any even positive integer n, Let | | p be the normalized absolute value of C p , such that |p| p = 1 p , and let Then, for each fixed t ∈ E, the function e zt is analytic on Z p and hence considered as a function on X, and, by applying (1.2) to f with f (z) = χ(z)e zt , we get the padic integral expression of the generating function for the generalized Euler numbers E n,χ attached to χ: So we have the following p-adic integral expression of the generating function for the generalized Euler polynomials E n,χ (x) attached to χ: Also, from (1.1) we have: Let T k (n, χ) denote the kth alternating generalized power sum of the first n + 1 nonnegative integers attached to χ, namely From (1.4), (1.6), and (1.7), one easily derives the following identities: for any odd positive integer w, In what follows, we will always assume that the p-adic integrals of the various (twisted) exponential functions on X are defined for t ∈ E (cf. (1.3)), and therefore it will not be mentioned.
[1], [2], [6], [9] and [10] are some of the previous works on identities of symmetry in two variables involving Bernoulli polynomials and power sums. On the other hand, for the first time we were able to produce in [4] some identities of symmetry in three variables related to Bernoulli polynomials and power sums and to extend in [3] to the case of generalized Bernoulli polynomials and generalized power sums. Also, [6] is about identities of symmetry in two variables for Euler polynomials and alternating power sums and [5] is about those in three variables for them.
In this paper, we will be able to produce 8 identities of symmetry in three variables regarding generalized Euler polynomials and alternating generalized power sums. The case of two variables was treated in [7].
The following is stated as Theorem 4.2 and an example of the full six symmetries in w 1 , w 2 , w 3 .
The derivations of identities are based on the p-adic integral expression of the generating function for the generalized Euler polynomials in (1.5) and the quotient of integrals in (1.8)-(1.10) that can be expressed as the exponential generating function for the alternating generalized power sums. These abundance of symmetries would not be unearthed if such p-adic integral representations had not been available. We indebted this idea to the paper [7].

Several types of quotients of p-adic fermionic integrals
Here we will introduce several types of quotients of p-adic fermionic integrals on X or X 3 from which some interesting identities follow owing to the built-in symmetries in w 1 , w 2 , w 3 . In the following, w 1 , w 2 , w 3 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 identities in (1.4) and (1.6). To ease notations, from now on we will suppress µ −1 and denote, for example, dµ −1 (x) simply by dx.

Identities for generalized Euler polynomials
In the following, w 1 , w 2 , w 3 are all odd positive integers except for (a-0) and (c-0), where they are any positive integers. First, let's consider Type Λ i 23 , for each i = 0, 1, 2, 3. The following results can be easily obtained from (1.5) and (1.8) where the inner sum is over all nonnegative integers k, l, m with k + l + m = n,and (a-1) Here we write I(Λ 1 23 ) in two different ways: t n n! . (3.4) (2) Invoking (1.9), (3.3) can also be written as t n n! .

Main theorems
As we noted earlier in the last paragraph of Section 2, the various types of quotients of p-adic fermionic integrals are invariant under any permutation of w 1 , w 2 , w 3 . So the corresponding expressions in Section 3 are also invariant under any permutation of w 1 , w 2 , w 3 . 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 w 1 , w 2 , w 3 in a single one labeled by them, they can be viewed as a group in a natural manner and hence it is isomorphic to a quotient of S 3 . In particular, the number of possible distinct expressions are 1, 2, 3, or 6 (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.
Here we will just consider the cases of Theorems 4.4 and 4.8, leaving the others as easy exercises for the reader. As for the case of Theorem 4.4, in addition to (4.11)-(4.13), we get the following three ones: But, by interchanging l and m, we see that (4.1), (4.2), and (4.3) are respectively equal to (4.11), (4.12), and (4.13). As to Theorem 4.8, in addition to (4.17) and (4.18), we have: However, (4.4) and (4.5) are equal to (4.17), as we can see by applying the permutations k → l, l → m, m → k for (4.4) and k → m, l → k, m → l for (4.5). Similarly, we see that (4.6) and (4.7) are equal to (4.18), by applying permutations k → l, l → m, m → k for (4.6) and k → m, l → k, m → l for (4.7). Theorem 4.1. Let w 1 , w 2 , w 3 be any positive integers. Then we have: (4.8) Theorem 4.2. Let w 1 , w 2 , w 3 be any odd positive integers. Then we have: