SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

On the generalized Apostol-type Frobenius-Euler polynomials

Burak Kurt1 and Yilmaz Simsek2*

Author Affiliations

1 Department of Mathematics, Faculty of Education, Akdeniz University, Antalya, 07058, Turkey

2 Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, 07058, Turkey

For all author emails, please log on.

Advances in Difference Equations 2013, 2013:1  doi:10.1186/1687-1847-2013-1

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/1


Received:8 November 2012
Accepted:13 December 2012
Published:4 January 2013

© 2013 Kurt and Simsek; licensee Springer

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.

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

Throughout this presentation, we use the following standard notions: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M1">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M2">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M3">View MathML</a>. 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, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M4">View MathML</a> and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M5">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M6">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M7">View MathML</a>.

The classical Frobenius-Euler polynomial <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M8">View MathML</a> of order α is defined by means of the following generating function:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M9">View MathML</a>

(1)

where u is an algebraic number and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M10">View MathML</a>.

Observe that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M11">View MathML</a>, which denotes the Frobenius-Euler polynomials and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M12">View MathML</a>, which denotes the Frobenius-Euler numbers of order α. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M13">View MathML</a>, which denotes the Euler polynomials (cf.[1-24]).

Definition 1.1 (for details, see [16,17])

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M14">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M15">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M16">View MathML</a>. The generalized Apostol-type Frobenius-Euler polynomials are defined by means of the following generating function:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M17">View MathML</a>

(2)

Remark 1.2 If we set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M18">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M19">View MathML</a> in (2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M20">View MathML</a>

(3)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M21">View MathML</a> 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 2.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M22">View MathML</a>. Each of the following relationships holds true:

(4)

(5)

(6)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M26">View MathML</a>

(7)

Proof of (6) From (2),

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M27">View MathML</a>

(8)

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M28">View MathML</a>

Thus, by using the Cauchy product in (8) and then equating the coefficients of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29">View MathML</a> 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

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M30">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M31">View MathML</a> is replaced by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M32">View MathML</a>.

Theorem 2.2Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M33">View MathML</a>. Then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M34">View MathML</a>

Proof By using (2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M35">View MathML</a>

By equating the coefficients of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29">View MathML</a> on both sides of the resulting equation, we obtain the desired result. □

Theorem 2.3The following relationship holds true:

(9)

Proof We set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M38">View MathML</a>

From the above equation, we see that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M39">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M40">View MathML</a>

Comparing the coefficients of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29">View MathML</a> on both sides of the above equation, we arrive at the desired result. □

Remark 2.4 By substituting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M42">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M43">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M44">View MathML</a> into Theorem 2.3, we get Carlitz’s results (for details, see [[1], Eq. 2.19]) as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M45">View MathML</a>

We give the following generating function of the polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M46">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M47">View MathML</a>

(10)

(cf.[16,17]). We also note that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M48">View MathML</a>

If we substitute <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M18">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M42">View MathML</a> into (10), we see that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M51">View MathML</a>

Theorem 2.5The generalized Apostol-type Frobenius-Euler polynomial holds true as follows:

(11)

Proof Substituting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M53">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M19">View MathML</a> into (2) and taking derivative with respect to t, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M55">View MathML</a>

Using (10), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M56">View MathML</a>

Thus, after some elementary calculations, we arrive at (11). □

Theorem 2.6Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M57">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M58">View MathML</a>. Then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M59">View MathML</a>

(12)

Proof In (2), we replace α by −α, then we set

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M60">View MathML</a>

By using (2), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M61">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M62">View MathML</a>

Comparing the coefficients of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29">View MathML</a> 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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M64">View MathML</a> is given as follows:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M65">View MathML</a>

(<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M66">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M67">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M68">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M69">View MathML</a> when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M70">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M71">View MathML</a>); <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M72">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M73">View MathML</a> when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M74">View MathML</a>). It is obvious that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M75">View MathML</a>

(13)

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M76">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M64">View MathML</a> 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 3.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M78">View MathML</a>. We have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M79">View MathML</a>

(14)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M80">View MathML</a>

Proof From (2), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M81">View MathML</a>

Therefore,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M82">View MathML</a>

Comparing the coefficients of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29">View MathML</a> on both sides of the above equation, we have arrive at (14). □

Remark 3.2 By substituting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M42">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M43">View MathML</a> into (14), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M86">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M87">View MathML</a>

Remark 3.3 The function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M88">View MathML</a> 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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M88">View MathML</a> for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M90">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M91">View MathML</a>.

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 <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M92">View MathML</a> by means of the following generating function:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M93">View MathML</a>

(15)

The generating function for these polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M94">View MathML</a> is given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M95">View MathML</a>

(16)

(cf.[17]).

The generalized Apostol-Bernoulli polynomials were defined by Srivastava et al. [[22], p.254, Eq. (20)] as follows.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M96">View MathML</a> with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M15">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M98">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M99">View MathML</a>. Then the generalized Bernoulli polynomials <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M100">View MathML</a> of order <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M10">View MathML</a> are defined by means of the following generating functions:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M102">View MathML</a>

(17)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M103">View MathML</a>

We note that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M104">View MathML</a> and also <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M105">View MathML</a>, which denotes the Apostol-Bernoulli polynomials (cf.[1-24]).

Theorem 4.1Letvbe an integer. Then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M106">View MathML</a>

Proof Replacing c by b in (2) and after some calculations, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M107">View MathML</a>

Comparing the coefficients of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29">View MathML</a> on both sides of the above equation, we arrive at the desired result. □

Corollary 4.2

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M109">View MathML</a>

Proof Replacing c by b in (2) and after some calculations, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M110">View MathML</a>

Comparing the coefficients of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M29">View MathML</a> 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

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

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

  3. Choi, J, Srivastava, HM: The multiple Hurwitz-Lerch zeta function and the multiple Hurwitz-Euler eta function. Taiwan. J. Math.. 15, 501–522 (2011)

  4. 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:10.1155/2012/861797

  5. Ding, D, Yang, J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math.. 20, 7–21 (2010)

  6. Garg, M, Jain, K, Srivastava, HM: Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions. Integral Transforms Spec. Funct.. 17, 803–815 (2006). Publisher Full Text OpenURL

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

  8. Kim, T: An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic p-adic invariant q-integrals on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M112">View MathML</a>. Rocky Mt. J. Math.. 41, 239–247 (2011). Publisher Full Text OpenURL

  9. Kim, T: Identities involving Frobenius-Euler polynomials arising from non-linear differential equations. J. Number Theory. 132, 2854–2865 (2012) arXiv:1201.5088v1

    arXiv:1201.5088v1

    Publisher Full Text OpenURL

  10. Kim, T, Choi, J: A note on the product of Frobenius-Euler polynomials arising from the p-adic integral on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/1/mathml/M114">View MathML</a>. Adv. Stud. Contemp. Math.. 22, 215–223 (2012)

  11. Kurt, B, Simsek, Y: Frobenious-Euler type polynomials related to Hermite-Bernoulli polynomials. AIP Conf. Proc.. 1389, 385–388 (2011)

  12. Lin, S-D, Srivastava, HM, Wang, P-Y: Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions. Integral Transforms Spec. Funct.. 17, 817–827 (2006). Publisher Full Text OpenURL

  13. Luo, Q-M: q-analogues of some results for the Apostol-Euler polynomials. Adv. Stud. Contemp. Math.. 20, 103–113 (2010)

  14. Luo, Q-M, Srivastava, HM: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput.. 217, 5702–5728 (2011). Publisher Full Text OpenURL

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

  16. Simsek, Y: Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials. Axioms. 1, 395–403 (2012) doi:10.3390/axioms1030395

    doi:10.3390/axioms1030395

    Publisher Full Text OpenURL

  17. Simsek, Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their application. arXiv:1111.3848v2

  18. 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.. 4, 519–529 (2004)

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

  20. 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.. 12, 241–268 (2005)

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

  22. 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 OpenURL

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

  24. Srivastava, HM, Choi, J: Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, Amsterdam (2012)