Let
be a fixed odd prime number. Throughout this paper,
and
will, respectively, denote the ring of
-adic rational integer, the field of
-adic rational numbers, the complex number field, and the completion of algebraic closure of
. Let
be the normalized exponential valuation of
with
. Let
be the space of uniformly differentiable functions on
. For
,
with
, the fermionic
-adic
-integral on
is defined as
(see [1]). Let us define the fermionic
-adic invariant integral on
as follows:
(see [1–8]). From (1.2), we have
(see [9, 10]), where
. For
with
, let
. Then, we define the
-Euler numbers as follows:
where
are called the
-Euler numbers. We can show that
where
are the Frobenius-Euler numbers. By comparing the coefficients on both sides of (1.4) and (1.5), we see that
Now, we also define the
-Euler polynomials as follows:
In the viewpoint of (1.5), we can show that
where
are the
th Frobenius-Euler polynomials. From (1.7) and (1.8), we note that
(cf. [1–8, 11–18]). For each positive integer
, let
. Then we have
The
-Euler polynomials of order
, denoted
, are defined as
Then the values of
at
are called the
-Euler numbers of order
. When
, the polynomials or numbers are called the
-Euler polynomials or numbers. The purpose of this paper is to investigate some properties of symmetry for the multivariate
-adic fermionic integral on
. From the properties of symmetry for the multivariate
-adic fermionic integral on
, we derive some identities of symmetry for the
-Euler polynomials of higher order. By using our identities of symmetry for the
-Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler polynomials of higher order.