- Research
- Open Access
On the generalized Apostol-type Frobenius-Euler polynomials
- Burak Kurt1 and
- Yilmaz Simsek2Email author
https://doi.org/10.1186/1687-1847-2013-1
© Kurt and Simsek; licensee Springer 2013
- Received: 8 November 2012
- Accepted: 13 December 2012
- Published: 4 January 2013
Abstract
The aim of this paper is to derive some new identities related to the Frobenius-Euler polynomials. We also give relation between the generalized Frobenius-Euler polynomials and the generalized Hurwitz-Lerch zeta function at negative integers. Furthermore, our results give generalized Carliz’s results which are associated with Frobenius-Euler polynomials.
MSC:05A10, 11B65, 28B99, 11B68.
Keywords
- Frobenius-Euler polynomials
- Hermite-based Frobenius-Euler polynomials
- Hermite-based Apostol-Euler polynomials
- Apostol-Euler polynomials
- Hurwitz-Lerch zeta function
1 Introduction, definitions and notations
where , .
where u is an algebraic number and .
Observe that , which denotes the Frobenius-Euler polynomials and , which denotes the Frobenius-Euler numbers of order α. , which denotes the Euler polynomials (cf. [1–24]).
Definition 1.1 (for details, see [16, 17])
where denotes the generalized Apostol-type Frobenius-Euler numbers (cf. [17]).
2 New identities
In this section, we derive many new identities related to the generalized Apostol-type Frobenius-Euler numbers and polynomials of order α.
Thus, by using the Cauchy product in (8) and then equating the coefficients of on both sides of the resulting equation, we obtain the desired result.
The proofs of (4), (5) and (7) are the same as that of (2), thus we omit them. □
where is replaced by .
By equating the coefficients of on both sides of the resulting equation, we obtain the desired result. □
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Thus, after some elementary calculations, we arrive at (11). □
Comparing the coefficients of on both sides of the above equation, we arrive at (12). □
3 Interpolation function
where denotes the Lerch-Zeta function (cf. [6, 19, 21, 24]).
Relation between the generalized Apostol-type Frobenius-Euler polynomials and the Hurwitz-Lerch zeta function is given as follows.
Comparing the coefficients of on both sides of the above equation, we have arrive at (14). □
Remark 3.3 The function is an interpolation function of the generalized Apostol-type Frobenius-Euler polynomials of order α at negative integers, which is given by the analytic continuation of the for , .
4 Relations between Array-type polynomials, Apostol-Bernoulli polynomials and generalized Apostol-type Frobenius-Euler polynomial
(cf. [17]).
The generalized Apostol-Bernoulli polynomials were defined by Srivastava et al. [[22], p.254, Eq. (20)] as follows.
We note that and also , which denotes the Apostol-Bernoulli polynomials (cf. [1–24]).
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Declarations
Acknowledgements
Dedicated to Professor Hari M. Srivastava.
All authors are partially supported by Research Project Offices Akdeniz Universities.
Authors’ Affiliations
References
- Carlitz L: Eulerian numbers and polynomials. Math. Mag. 1959, 32: 247–260. 10.2307/3029225MathSciNetView ArticleGoogle Scholar
- Choi J, Jang SD, Srivastava HM: A generalization of the Hurwitz-Lerch zeta function. Integral Transforms Spec. Funct. 2008, 19: 65–79.MathSciNetView ArticleGoogle Scholar
- Choi J, Srivastava HM: The multiple Hurwitz-Lerch zeta function and the multiple Hurwitz-Euler eta function. Taiwan. J. Math. 2011, 15: 501–522.MathSciNetGoogle Scholar
- Choi J, Kim DS, Kim T, Kim YH: A note on some identities of Frobenius-Euler numbers and polynomials. Int. J. Math. Math. Sci. 2012. doi:10.1155/2012/861797Google Scholar
- Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20: 7–21.MathSciNetGoogle Scholar
- Garg M, Jain K, Srivastava HM: Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions. Integral Transforms Spec. Funct. 2006, 17: 803–815. 10.1080/10652460600926907MathSciNetView ArticleGoogle Scholar
- Gould HW: The q -series generalization of a formula of Sparre Andersen. Math. Scand. 1961, 9: 90–94.MathSciNetGoogle Scholar
- Kim T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p -adic invariant q -integrals on. Rocky Mt. J. Math. 2011, 41: 239–247. 10.1216/RMJ-2011-41-1-239View ArticleGoogle Scholar
- Kim T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory 2012, 132: 2854–2865. arXiv:1201.5088v1 10.1016/j.jnt.2012.05.033MathSciNetView ArticleGoogle Scholar
- Kim T, Choi J: A note on the product of Frobenius-Euler polynomials arising from the p -adic integral on. Adv. Stud. Contemp. Math. 2012, 22: 215–223.Google Scholar
- Kurt B, Simsek Y: Frobenious-Euler type polynomials related to Hermite-Bernoulli polynomials. AIP Conf. Proc. 2011, 1389: 385–388.View ArticleGoogle Scholar
- Lin S-D, Srivastava HM, Wang P-Y: Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions. Integral Transforms Spec. Funct. 2006, 17: 817–827. 10.1080/10652460600926923MathSciNetView ArticleGoogle Scholar
- Luo Q-M: q -analogues of some results for the Apostol-Euler polynomials. Adv. Stud. Contemp. Math. 2010, 20: 103–113.Google Scholar
- Luo Q-M, Srivastava HM: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 2011, 217: 5702–5728. 10.1016/j.amc.2010.12.048MathSciNetView ArticleGoogle Scholar
- Srivastava HM, Saxena RK, Pogany TK, Saxena R: Integral and computational representation of the extended Hurwitz-Lerch zeta function. Integral Transforms Spec. Funct. 2011, 22: 487–506. 10.1080/10652469.2010.530128MathSciNetView ArticleGoogle Scholar
- Simsek Y: Generating functions for q -Apostol type Frobenius-Euler numbers and polynomials. Axioms 2012, 1: 395–403. doi:10.3390/axioms1030395 10.3390/axioms1030395View ArticleGoogle Scholar
- Simsek, Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their application. arXiv:1111.3848v2Google Scholar
- Simsek Y, Kim T, Park DW, Ro YS, Jang LC, Rim SH: An explicit formula for the multiple Frobenius-Euler numbers and polynomials. JP J. Algebra Number Theory Appl. 2004, 4: 519–529.MathSciNetGoogle Scholar
- Srivastava HM: Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inform. Sci. 2011, 5: 390–444.Google Scholar
- Srivastava HM, Kim T, Simsek Y: q -Bernoulli numbers and polynomials associated with multiple q -zeta functions and basic L -series. Russ. J. Math. Phys. 2005, 12: 241–268.MathSciNetGoogle Scholar
- Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht; 2001.View ArticleGoogle Scholar
- Srivastava HM, Garg M, Choudhary S: A new generalization of the Bernoulli and related polynomials. Russ. J. Math. Phys. 2010, 17: 251–261. 10.1134/S1061920810020093MathSciNetView ArticleGoogle Scholar
- Srivastava HM, Garg M, Choudhary S: Some new families of the generalized Euler and Genocchi polynomials. Taiwan. J. Math. 2011, 15: 283–305.MathSciNetGoogle Scholar
- Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.