Abstract
The aim of this paper is to derive some new identities related to the FrobeniusEuler polynomials. We also give relation between the generalized FrobeniusEuler polynomials and the generalized HurwitzLerch zeta function at negative integers. Furthermore, our results give generalized Carliz’s results which are associated with FrobeniusEuler polynomials.
MSC: 05A10, 11B65, 28B99, 11B68.
Keywords:
FrobeniusEuler polynomials; Hermitebased FrobeniusEuler polynomials; Hermitebased ApostolEuler polynomials; ApostolEuler polynomials; HurwitzLerch zeta function1 Introduction, definitions and notations
Throughout this presentation, we use the following standard notions: , , . Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real numbers and ℂ denotes the set of complex numbers. Furthermore, and
The classical FrobeniusEuler polynomial of order α is defined by means of the following generating function:
where u is an algebraic number and .
Observe that , which denotes the FrobeniusEuler polynomials and , which denotes the FrobeniusEuler numbers of order α. , which denotes the Euler polynomials (cf.[124]).
Definition 1.1 (for details, see [16,17])
Let , , . The generalized Apostoltype FrobeniusEuler polynomials are defined by means of the following generating function:
Remark 1.2 If we set and in (2), we get
where denotes the generalized Apostoltype FrobeniusEuler numbers (cf.[17]).
2 New identities
In this section, we derive many new identities related to the generalized Apostoltype FrobeniusEuler numbers and polynomials of order α.
Theorem 2.1Let. Each of the following relationships holds true:
and
Proof of (6) From (2),
Therefore,
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. □
Observe that in (6) we have
Proof By using (2), we get
By equating the coefficients of on both sides of the resulting equation, we obtain the desired result. □
Theorem 2.3The following relationship holds true:
Proof We set
From the above equation, we see that
Therefore,
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Remark 2.4 By substituting , , into Theorem 2.3, we get Carlitz’s results (for details, see [[1], Eq. 2.19]) as follows:
We give the following generating function of the polynomials :
(cf.[16,17]). We also note that
If we substitute and into (10), we see that
Theorem 2.5The generalized Apostoltype FrobeniusEuler polynomial holds true as follows:
Proof Substituting for into (2) and taking derivative with respect to t, we obtain
Using (10), we have
Thus, after some elementary calculations, we arrive at (11). □
Theorem 2.6Letand. Then we have
Proof In (2), we replace α by −α, then we set
By using (2), we get
Therefore,
Comparing the coefficients of on both sides of the above equation, we arrive at (12). □
3 Interpolation function
In this section, we give a recurrence relation between the generalized FrobeniusEuler polynomials and the HurwitzLerch zeta function. Recently, many authors have studied not only the HurwitzLerch zeta function, but also its generalizations, for example (among others), Srivastava [19], Srivastava and Choi [24] and also Garg et al.[6]. The generalization of the HurwitzLerch zeta function is given as follows:
(, , , when (); and when ). It is obvious that
and
where denotes the LerchZeta function (cf.[6,19,21,24]).
Relation between the generalized Apostoltype FrobeniusEuler polynomials and the HurwitzLerch zeta function is given as follows.
where
Proof From (2), we have
Therefore,
Comparing the coefficients of on both sides of the above equation, we have arrive at (14). □
Remark 3.2 By substituting , into (14), we have
where
Remark 3.3 The function is an interpolation function of the generalized Apostoltype FrobeniusEuler polynomials of order α at negative integers, which is given by the analytic continuation of the for , .
4 Relations between Arraytype polynomials, ApostolBernoulli polynomials and generalized Apostoltype FrobeniusEuler polynomial
In [17], Simsek constructed the generalized λStirling type numbers of the second kind by means of the following generating function:
The generating function for these polynomials is given by
(cf.[17]).
The generalized ApostolBernoulli polynomials were defined by Srivastava et al. [[22], p.254, Eq. (20)] as follows.
Let with , and . Then the generalized Bernoulli polynomials of order are defined by means of the following generating functions:
where
We note that and also , which denotes the ApostolBernoulli polynomials (cf.[124]).
Theorem 4.1Letvbe an integer. Then we have
Proof Replacing c by b in (2) and after some calculations, we have
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Corollary 4.2
Proof Replacing c by b in (2) and after some calculations, we have
Comparing the coefficients of on both sides of the above equation, we arrive at the desired result. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M. Srivastava.
All authors are partially supported by Research Project Offices Akdeniz Universities.
References

Carlitz, L: Eulerian numbers and polynomials. Math. Mag.. 32, 247–260 (1959). Publisher Full Text

Choi, J, Jang, SD, Srivastava, HM: A generalization of the HurwitzLerch zeta function. Integral Transforms Spec. Funct.. 19, 65–79 (2008)

Choi, J, Srivastava, HM: The multiple HurwitzLerch zeta function and the multiple HurwitzEuler eta function. Taiwan. J. Math.. 15, 501–522 (2011)

Choi, J, Kim, DS, Kim, T, Kim, YH: A note on some identities of FrobeniusEuler numbers and polynomials. Int. J. Math. Math. Sci. (2012) doi:10.1155/2012/861797

Ding, D, Yang, J: Some identities related to the ApostolEuler and ApostolBernoulli polynomials. Adv. Stud. Contemp. Math.. 20, 7–21 (2010)

Garg, M, Jain, K, Srivastava, HM: Some relationships between the generalized ApostolBernoulli polynomials and HurwitzLerch zeta functions. Integral Transforms Spec. Funct.. 17, 803–815 (2006). Publisher Full Text

Gould, HW: The qseries generalization of a formula of Sparre Andersen. Math. Scand.. 9, 90–94 (1961)

Kim, T: An identity of the symmetry for the FrobeniusEuler polynomials associated with the fermionic padic invariant qintegrals on . Rocky Mt. J. Math.. 41, 239–247 (2011). Publisher Full Text

Kim, T: Identities involving FrobeniusEuler polynomials arising from nonlinear differential equations. J. Number Theory. 132, 2854–2865 (2012) arXiv:1201.5088v1
arXiv:1201.5088v1
Publisher Full Text 
Kim, T, Choi, J: A note on the product of FrobeniusEuler polynomials arising from the padic integral on . Adv. Stud. Contemp. Math.. 22, 215–223 (2012)

Kurt, B, Simsek, Y: FrobeniousEuler type polynomials related to HermiteBernoulli polynomials. AIP Conf. Proc.. 1389, 385–388 (2011)

Lin, SD, Srivastava, HM, Wang, PY: Some expansion formulas for a class of generalized HurwitzLerch zeta functions. Integral Transforms Spec. Funct.. 17, 817–827 (2006). Publisher Full Text

Luo, QM: qanalogues of some results for the ApostolEuler polynomials. Adv. Stud. Contemp. Math.. 20, 103–113 (2010)

Luo, QM, Srivastava, HM: Some generalizations of the ApostolGenocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput.. 217, 5702–5728 (2011). Publisher Full Text

Srivastava, HM, Saxena, RK, Pogany, TK, Saxena, R: Integral and computational representation of the extended HurwitzLerch zeta function. Integral Transforms Spec. Funct.. 22, 487–506 (2011). Publisher Full Text

Simsek, Y: Generating functions for qApostol type FrobeniusEuler numbers and polynomials. Axioms. 1, 395–403 (2012) doi:10.3390/axioms1030395
doi:10.3390/axioms1030395
Publisher Full Text 
Simsek, Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their application. arXiv:1111.3848v2

Simsek, Y, Kim, T, Park, DW, Ro, YS, Jang, LC, Rim, SH: An explicit formula for the multiple FrobeniusEuler numbers and polynomials. JP J. Algebra Number Theory Appl.. 4, 519–529 (2004)

Srivastava, HM: Some generalizations and basic (or q) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inform. Sci.. 5, 390–444 (2011)

Srivastava, HM, Kim, T, Simsek, Y: qBernoulli numbers and polynomials associated with multiple qzeta functions and basic Lseries. Russ. J. Math. Phys.. 12, 241–268 (2005)

Srivastava, HM, Choi, J: Series Associated with the Zeta and Related Functions, Kluwer Academic, Dordrecht (2001)

Srivastava, HM, Garg, M, Choudhary, S: A new generalization of the Bernoulli and related polynomials. Russ. J. Math. Phys.. 17, 251–261 (2010). Publisher Full Text

Srivastava, HM, Garg, M, Choudhary, S: Some new families of the generalized Euler and Genocchi polynomials. Taiwan. J. Math.. 15, 283–305 (2011)

Srivastava, HM, Choi, J: Zeta and qZeta Functions and Associated Series and Integrals, Elsevier, Amsterdam (2012)