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.
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1 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
where , .
The classical Frobenius-Euler 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 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])
Let , , . The generalized Apostol-type Frobenius-Euler 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 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.
Theorem 2.1 Let . 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
where is replaced by .
Theorem 2.2 Let . Then 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 2.3 The 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 2.5 The generalized Apostol-type Frobenius-Euler 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 2.6 Let and . 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 Frobenius-Euler polynomials and the Hurwitz-Lerch zeta function. Recently, many authors have studied not only the Hurwitz-Lerch 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 Hurwitz-Lerch zeta function is given as follows:
(, , , when (); and when ). It is obvious that
and
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.
Theorem 3.1 Let . We have
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 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
In [17], Simsek constructed the generalized λ-Stirling 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 Apostol-Bernoulli 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 Apostol-Bernoulli polynomials (cf. [1–24]).
Theorem 4.1 Let v be 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. □
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Acknowledgements
Dedicated to Professor Hari M. Srivastava.
All authors are partially supported by Research Project Offices Akdeniz Universities.
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Kurt, B., Simsek, Y. On the generalized Apostol-type Frobenius-Euler polynomials. Adv Differ Equ 2013, 1 (2013). https://doi.org/10.1186/1687-1847-2013-1
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DOI: https://doi.org/10.1186/1687-1847-2013-1