**Turkish Journal of Analysis and Number Theory**

##
Symmetric Identities Involving *q*-Frobenius-Euler Polynomials under Sym (5)

**Serkan Araci**^{1,}, **Ugur Duran**^{2}, **Mehmet Acikgoz**^{2}

^{1}Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, Gaziantep, TURKEY

^{2}University of Gaziantep, Faculty of Science and Arts, Department of Mathematics, Gaziantep, TURKEY

Abstract | |

1. | Introduction |

2. | Symmetric Identities Involving q-Frobenius-Euler Polynomials |

3. | Conclusion |

References |

### Abstract

Following the definition of *q*-Frobenius-Euler polynomials introduced in [3], we derive some new symmetric identities under sym (5), also termed symmetric group of degree five, which are derived from the fermionic *p-*adic *q-*integral over the *p-*adic numbers field.

**Keywords:** Symmetric identities, q-Frobenius-Euler polynomials, Fermionic p-adic q-integral on Invariant under S_{5}

Received March 09, 2015; Revised May 23, 2015; Accepted August 28, 2015

**Copyright**© 2015 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Serkan Araci, Ugur Duran, Mehmet Acikgoz. Symmetric Identities Involving
*q*-Frobenius-Euler Polynomials under Sym (5).*Turkish Journal of Analysis and Number Theory*. Vol. 3, No. 3, 2015, pp 90-93. http://pubs.sciepub.com/tjant/3/3/5

- Araci, Serkan, Ugur Duran, and Mehmet Acikgoz. "Symmetric Identities Involving
*q*-Frobenius-Euler Polynomials under Sym (5)."*Turkish Journal of Analysis and Number Theory*3.3 (2015): 90-93.

- Araci, S. , Duran, U. , & Acikgoz, M. (2015). Symmetric Identities Involving
*q*-Frobenius-Euler Polynomials under Sym (5).*Turkish Journal of Analysis and Number Theory*,*3*(3), 90-93.

- Araci, Serkan, Ugur Duran, and Mehmet Acikgoz. "Symmetric Identities Involving
*q*-Frobenius-Euler Polynomials under Sym (5)."*Turkish Journal of Analysis and Number Theory*3, no. 3 (2015): 90-93.

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### 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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