1. Introduction

As it is known, the Frobenius-Euler polynomials for with are defined by means of the power series expansion at

(1.1)

Taking x = 0 in the Eq. (1.1), we have that is widely known as n-th Frobenius-Euler number cf. ^{[3, 4, 5, 8, 17, 18, 21]}.

Let p be chosen as a fixed odd prime number. Throughout this paper, we make use of the following notations: denotes topological closure of , denotes the field of rational numbers, denotes topological closure of , and indicates the field of p-adic completion of an algebraic closure of . Let be the set of natural numbers and

For d an odd positive number with (p,d) = 1, let

and

where lies in . See, for details, [1,2,3,4,6-17].

The normalized absolute value according to the theory of p-adic analysis is given by . q can be considered as an indeterminate a complex number with , or a p-adic number with and for It is always clear in the content of the paper.

Throughout this paper, we use the following notation:

(1.2)

which is called q-extension of x. It easily follows that for any x.

Let f be uniformly differentiable function at a point which is denoted by Then the p-adic q-integral on (or sometimes called q-Volkenborn integral) of a function f is defined by Kim ^[10]

(1.3)

It follows from the Eq. (1.3) that

(1.4)

Thus, by the Eq. (1.4), we have

where . For the applications of fermionic p-adic integral over the p-adic numbers field, see the references, e. g., ^{[1, 2, 3, 4, 6, 7, 9, 11, 12, 16]}.

In ^[3], the q-Frobenius-Euler polynomials are defined by the following p-adic fermionic q-integral on , with respect to :

(1.5)

Upon setting x = 0 into the Eq. (1.5) gives which are called n-th q-Frobenius-Euler number.

By letting in the Eq. (1.5), it yields to

Recently, many mathematicians have studied the symmetric identities on some special polynomials, see, for details, ^{[1, 6, 7, 9, 12]}. Some of mathematicians also investigated some applications of Frobenius-Euler numbers and polynomials (or its q-analog) cf. ^{[3, 4, 5, 13, 14, 15, 16]}. Moreover, Frobenius-Euler numbers at the value λ = −1 in (1.1) are Euler numbers that is closely related to Bernoulli numbers, Genocchi numbers, etc. For more information about these polynomials, look at [1-21]^[1] and the references cited therein.

In the present paper, we obtain not only new but also some interesting identities which are derived from the fermionic p-adic q-integral over the p-adic numbers field. The results derived here is written under Sym (5).

2. Symmetric Identities Involving q-Frobenius-Euler Polynomials

For with with by the Eqs. (1.3) and (1.5), we obtain

(2.1)

Taking

on the both sides of Eq. (2.1) gives

(2.2)

Note that the equation (2.2) is invariant for any permutation Hence, we have the following theorem.

Theorem 1. Let with with Then the following

holds true for any

By Eq. (1.2), we easily derive that

(2.3)

From Eq. (2.1) and (2.3), we obtain

(2.4)

from which, we have

(2.5)

Thus, by Theorem 1 and (2.5), we procure the following theorem.

Theorem 2. For with with the following

holds true for any

It is shown by using the definition of that

(2.6)

Taking on the both sides of Eq.(2.6) gives

(2.7)

By the Eq. (2.7), we have

(2.8)

where

(2.9)

Consequently, by (2.9), we get the following theorem.

Theorem 3. Let with with Then the following expression

holds true for some

3. Conclusion

We have derived some new interesting identities of q-Frobenius-Euler polynomials. We also showed that these symmetric identities are written by symmetric group of degree five.

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