Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

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

Serkan Araci*, Ugur Duran, Mehmet Acikgoz

Department of Economics, Faculty of Economics, Administrative and Social Sciences, Hasan Kalyoncu University, Gaziantep, TURKEY

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 p Invariant under S₅

Citation: Serkan Araci, Ugur Duran, Mehmet Acikgoz, Symmetric Identities Involving q-Frobenius-Euler Polynomials under Sym (5). Turkish Journal of Analysis and Number Theory, 3 (3). 90-93. January 2015. doi:10.12691/tjant-3-3-5

1. Introduction

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

[picture] | (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:

[picture] | (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]

[picture] | (1.3)

It follows from the Eq. (1.3) that

[picture] | (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 :

[picture] | (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] 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

[picture] | (2.1)

Taking

on the both sides of Eq. (2.1) gives

[picture] | (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

[picture] | (2.3)

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

[picture] | (2.4)

from which, we have

[picture] | (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

[picture] | (2.6)

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

[picture] | (2.7)

By the Eq. (2.7), we have

[picture] | (2.8)

where

[picture] | (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.

References

[1]   | E. Ağyüz, M. Acikgoz and S. Araci, A symmetric identity on the q-Genocchi polynomials of higher order under third Dihedral group D₃, Proc. Jangjeon Math. Soc. 18 (2015), No. 2, pp. 177-187.
[2]   | S. Araci, M. Acikgoz, E. ¸Sen, On the extended Kim.s p-adic q-deformed fermionic integrals in the p-adic integer ring, J. Number Theory 133 (2013) 3348-3361.
[3]   | S. Araci, M. Acikgoz, On the von Staudt-Clausen.s theorem related to q-Frobenius-Euler number, J. Number Theory (2016).
[4]   | S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. 22 (2012), No. 3, pp. 399-406.
[5]   | S. Araci, E. ¸Sen, M. Acikgoz, Theorems on Genocchi polynomials of higher order arising from Genocchi basis, Taiwanese J. Math.(2014) Vol. 18, No. 2, pp. 473-482.
[6]   | D. V. Dolgy, Y. S. Jang, T. Kim, H. I. Kwon, J.-J. Seo, Identities of symmetry for q-Euler polynomials derived from fermionic integral onp under symmetry group S₃, Applied Mathematical Sciences, Vol. 8 (2014), no. 113, 5599-5607.
[7]   | D. V. Dolgy, T. Kim, S.-H. Rim, S.-H. Lee, Some symmetric identities for h-extension of q-Euler polynomials under third dihedral group D₃, International Journal of Mathematical Analysis Vol. 8 (2014), no. 48, 2369-2374.
[8]   | Y. He and S. J. Wang, New formulae of products of the Frobenius-Euler polynomials, J. Ineq. Appl. (2014), 2014:261.
[9]   | Y. S. Jang, T. Kim, S.-H. Rim, J.-J. Seo, Symmetry Identities for the Generalized Higher-Order q-Bernoulli Polynomials under S₃, International J. Math. Anal., Vol. 8 (2014), no. 38, 1873-1879.
[10]   | T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9 (2002), no. 3, 288-299.
[11]   | T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on p, Russian J. Math. Phys., 16, 484-491 (2009).
[12]   | T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, Journal of Nonlinear Mathematical Physics 14 (1) 15-27, (2007).
[13]   | T. Kim and J. J. Seo, New identities of symmetry for Carlitz.s-type q-Bernoulli polynomials under symmetric group of degree five, International Journal of Mathematical Analysis Vol. 9, 2015, no. 35, 1707-1713.
[14]   | T. Kim and J. J. Seo, Some identities of symmetry for Carlitz-type q-Euler polynomials invariant under symmetric group of degree five, International Journal of Mathematical Analysis Vol. 9, 2015, no. 37, 1815-1822.
[15]   | T. Kim, Some New identities of symmetry for higher-order Carlitz q-Bernoulli polynomials arising from p-adic q-integral on p under the symmetric group of degree five, Applied Mathematical Sciences, Vol. 9, 2015, no. 93, 4627-4634.
[16]   | D. S. Kim and T. Kim, Some identities of symmetry for Carlitz q-Bernoulli polynomials invariant under S₄, Ars Combinatoria, Vol. CXXIII, pp. 283-289, 2015.
[17]   | D. S. Kim and T. Kim, Some new identities of Frobenius-Euler numbers and polynomials, J. Ineq. Appl. (2012), 2012:307.
[18]   | Y. Simsek, Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials, Axioms (2012), 1, 395-403.
[19]   | H. M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Camb. Philos. Soc. 129, 77-84 (2000).
[20]   | H. M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inform. Sci. 5, 390-444 (2011).
[21]   | B. Y. Yasar and M. A. Özarslan, Frobenius-Euler and Frobenius-Genocchi polynomials and their di¤ erential equations, NTMSCI 3 (2015), No. 2, 172-180.