Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Go
Turkish Journal of Analysis and Number Theory. 2013, 1(1), 54-58
DOI: 10.12691/tjant-1-1-11
Open AccessResearch Article

Some Relationships between the Generalized Apostol-Bernoulli and Apostol-Euler Polynomials

Burak Kurt1,

1Department of Mathematical Education, Faculty of Educations, Akdeniz University, TR-07058 Antalya, Turkey

Pub. Date: December 03, 2013
(This article belongs to the Special Issue Recent developments in the areas of mathematics)

Cite this paper:
Burak Kurt. Some Relationships between the Generalized Apostol-Bernoulli and Apostol-Euler Polynomials. Turkish Journal of Analysis and Number Theory. 2013; 1(1):54-58. doi: 10.12691/tjant-1-1-11

Abstract

The main objective of this paper is to introduce and investigate two new classes of generalized Apostol-Bernoulli polynomials Bn[m-1,α](x;c,α;λ) and Apostol-Euler polynomials εn[m-1,α](x;c,α;λ). In particular, we obtain addition formula for the new class of the generalized Apostol-Bernoulli polynomials. We also give some recurrence relations and Raabe relations for these polynomials.

Keywords:
Bernoulli polynomials and numbers Apostol-Bernoulli polynomials Apostol-Euler polynomials Generalized Apostol-Bernoulli polynomials

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  A. Bagdasaryan and S. Araci, Some new identities on the Apostol-Bernoulli polynomials higher order derived from Bernoulli basis, arXiv:1311.4148 [math.NT].
 
[2]  G. Bretti and P. E. Ricci, Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese J. of Math. 8, 415-428, 2004.
 
[3]  S. Chen, Yi Chai and Q.-M. Luo, An extension of generalized Apostol-Euler polynomials, Advances in Difference Equation.
 
[4]  F. Costabile, F. Dellaccio and M. I. Gualtieri, A new approach to Bernoulli polynomials, Rendi. di. Math. Series VII, 26, 1-12, 2006.
 
[5]  S. Gaboury and B. Kurt, Some relations involving Hermite-based Apostol-Genocchi polynomials, App. Math. Sci., 82, 4091-4102, 2012.
 
[6]  Y. He and C. Wang, Some formulae of products of the Apostol-Bernoulli and Apostol-Euler polynomials, Discrete Dynamics in Nature and Society, Article ID 927953, 11 pages, 2012.
 
[7]  T. Kim, Some identities for the Bernoulli, Euler and Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. 20, 18-23, 2010.
 
[8]  T. Kim, T. Mansour, S.-H. Rim and S.-H. Lee, Apostol-Euler polynomials arising from umbral calculus, Advances in Difference Equations 2013, 2013:300.
 
[9]  B. Kurt, A further generalization of the Bernoulli polynomials and on the 2D-Bernoulli polynomials Bn2(x,y), App. Math. Sci., 47, 2315-2322, 2010.
 
[10]  B. Kurt, A further generalization of the Euler polynomials and on the 2D-Euler polynomials, Proc. Jang. Math. Soc., 15, 389-394, 2012.
 
[11]  Q.-M. Luo, The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order, Int. Trans. Spec. Func. Vol 20(5), 377-391, 2009.
 
[12]  P. Natalini and A. Bernardini, A Generalization of the Bernoulli polynomials, J. of App. Math., 153-163, 2003.
 
[13]  H. M. Srivastava and A. Pinter, Remarks on some relationships between the Bernoulli and Euler polynomials, App. Math. Letter, 17, 375-380, 2004.
 
[14]  H. M. Srivastava, M. Garg and S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russian J. of Math. Phys., 17, 251-261, 2010.
 
[15]  H. M. Srivastava, M. Garg and S. Choudhary, Some new families of neralized Euler and Genocchi polynomials, Taiwanese J. of Math., 15, 283-305, 2011.
 
[16]  R. Trembly, S. Gaboury and B.-J. Fugére, A new class of generalized Apostol-Bernoulli polynomials and some analogues of the Srivastava-Pinter addition theorem, Applied Math. Letter, 24, 1888-1893, 2011.
 
[17]  R. Trembly, S. Gaboury and B.-J. Fugére, Some new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials, Inter. J. of Math. and Math. Sci., 2012.