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

Open Access Research

Generating function for q-Eulerian polynomials and their decomposition and applications

Mustafa Alkan* and Yilmaz Simsek

Author Affiliations

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

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Fixed Point Theory and Applications 2013, 2013:72  doi:10.1186/1687-1812-2013-72

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/72


Received:28 December 2012
Accepted:6 March 2013
Published:27 March 2013

© 2013 Alkan 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 define a generating function for q-Eulerian polynomials and numbers attached to any character χ of the finite cyclic group G. We derive many functional equations, q-difference equations and partial deferential equations related to these generating functions. By using these equations, we find many properties of q-Eulerian polynomials and numbers. Using the generating element of the finite cyclic group G and the generating element of the subgroups of G, we show that the generating function with a conductor f can be written as a sum of the generating function with conductors which are less than f.

MSC: 05A40, 11B83, 11B68, 11S80.

Keywords:
Euler numbers; Frobenius-Euler numbers; Frobenius-Euler polynomials; q-Frobenius-Euler polynomials; q-series; generating function; character χ of the finite abelian groups G

1 Introduction

The theory of the family of Eulerian polynomials and numbers (Frobenius-Euler polynomials and numbers) has become a very important part of analytic number theory and other sciences, for example, engineering, computer, geometric design and mathematical physics. Euler numbers are related to the Brouwer fixed point theorem and vector fields. Therefore, q-Eulerian type numbers may be related to Brouwer fixed point theorem and vector fields [1].

Recently, many different special functions have been used to define and investigate q-Eulerian numbers and polynomials, see details [1-39]. Therefore, we construct and investigate various properties of q-Eulerian numbers and polynomials, which are related to the many known polynomials and numbers such as Frobenius-Euler polynomials and numbers, Apostol-Euler polynomials and numbers, Euler polynomials and numbers.

Recently in [2-4], the authors defined a relationship between algebra and analysis. In detail, they made a new approximation between the subgroup and monoid presentation and special generating functions (such as Stirling numbers, Array polynomials etc.). In this paper, since our priority aim is to define special numbers and polynomials, it is worth depicting these references as well. Then in this paper, applying any group character χ of the finite cyclic group G to a special generating function (which has been defined in [5]), we give a generalization of q-Eulerian polynomials and numbers (q-Apostol-typeFrobenius-Euler polynomials and numbers) and investigate their properties and some useful functional equations. Using a generating element of the subgroups of G and a generating element of G, we also decompose our generating function attached to the characters of G, and so we obtain a new decomposition of q-Eulerian numbers and polynomials attached to the characters of a subgroup of G and the Dirichlet character of G. This decomposition enables us to compute q-Apostol-type Frobenius-Euler polynomials and numbers more easily.

Throughout this paper, we assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M1">View MathML</a>, the set of complex numbers, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M2">View MathML</a>

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M3">View MathML</a>

We use the following standard notions.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M5">View MathML</a> and also, as usual, ℝ denotes the set of real numbers, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M6">View MathML</a> denotes the set of positive real numbers and ℂ denotes the set of complex numbers.

1.1 Characters of a group G

We recall the definition and some properties of the character of an arbitrary group <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M7">View MathML</a> (see [6]).

A non-zero complex-valued function χ defined on G is called a character of G if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M8">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M9">View MathML</a>

In particular, if G is a finite group with the identity element e, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M10">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M11">View MathML</a> is a root of unity. In [[6], Theorem 6.8], it is proved that a finite abelian group G of order n has exactly n distinct characters.

Let G be the group of reduced residue classes module positive integer f. Corresponding to each character χ of G, the Dirichlet character <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M12">View MathML</a> is defined as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M13">View MathML</a>

Hence it is easily observed that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M14">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M15">View MathML</a>. In this note, f is called the conductor of the character of a group G of reduced residue classes module a positive integer f.

1.2 q-Eulerian polynomials and numbers

In [5], Simsek defined and studied some properties of q-Apostol type Frobenius-Euler polynomials and numbers.

Definition 1.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M16">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M17">View MathML</a>) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M18">View MathML</a>.

(i) The q-Apostol type Frobenius-Euler numbers

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M19">View MathML</a>

are defined by means of the following generating function:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M20">View MathML</a>

(ii) The q-Apostol type Frobenius-Euler polynomials

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M21">View MathML</a>

are defined by means of the following generating function:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M22">View MathML</a>

(1)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M23">View MathML</a>

2 q-Eulerian polynomials and numbers attached to any character

In this section, we define q-Eulerian polynomials and numbers attached to any character of the finite cyclic group G. Our new generating functions are related to nonnegative real parameters.

Definition 2.1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M16">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M17">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M26">View MathML</a>), <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M27">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M28">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M29">View MathML</a>). Let χ be a character of a finite cyclic group G with the conductor f.

(i) The q-Eulerian numbers attached to the character χ

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M30">View MathML</a>

are defined by means of the following generating function:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M31">View MathML</a>

(2)

(ii) The q-Eulerian polynomials attached to the character χ

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M32">View MathML</a>

are defined by means of the following generating function:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M33">View MathML</a>

(3)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M34">View MathML</a>

It is observed that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M35">View MathML</a>

Upon setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M36">View MathML</a> in (3), we compute a q-Eulerian number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M37">View MathML</a> attached to the character χ as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M38">View MathML</a>

By using the conductor f of the character χ and combining with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M39">View MathML</a>, we modify Equation (2) and Equation (3), respectively, as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M40">View MathML</a>

(4)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M41">View MathML</a>

(5)

Therefore, we provide the following relationships between q-Eulerian numbers and q-Eulerian numbers attached to the character χ.

Theorem 2.2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42">View MathML</a>. Then we have

(i)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M43">View MathML</a>

(ii)

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M44">View MathML</a>

Proof By using (4), we deduce that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M45">View MathML</a>

which, by comparing the coefficient <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M46">View MathML</a> on the both sides of the above equations, yields the first assertion of Theorem 2.2.

The second assertion (ii) is proved with the same argument. □

By Theorem 2.2, we also compute a q-Eulerian number attached to the character χ as follows:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M47">View MathML</a>

Now, we turn our attention to the following generation function defined in [1] since we need this generating function frequently to give some functional equations for a q-Eulerian number and polynomials attached to the character χ.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M26">View MathML</a>. The number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M49">View MathML</a> is defined by means of the following generating function:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M50">View MathML</a>

The polynomials <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M51">View MathML</a> are defined by means of the following generating function:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M52">View MathML</a>

(6)

Since we need this generating function frequently in this paper, we use the notation

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M53">View MathML</a>

and so it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M54">View MathML</a>.

By using the following well-known identity:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M55">View MathML</a>

in (3), we verify the following functional equation:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M56">View MathML</a>

(7)

Hence we have the following theorem.

Theorem 2.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M58">View MathML</a>

Proof By applying the Cauchy product to (7), we deduce that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M59">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M60">View MathML</a>. Then it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M61">View MathML</a>

By comparing the coefficient of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M62">View MathML</a> on both sides of the above equation, we obtain our desired result. □

Upon setting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M63">View MathML</a> in (7), we get the following functional equation:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M64">View MathML</a>

By substituting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M63">View MathML</a> in Theorem 2.3, the following theorem is easily proved.

Theorem 2.4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M67">View MathML</a>

So that we obtain a q-difference equation for q-Eulerian polynomials attached to the character χ, we study the following equations:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M68">View MathML</a>

which, in light of the Cauchy product of the three series <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M69">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M70">View MathML</a> and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M71">View MathML</a>

yield the following theorem.

Theorem 2.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M73">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M74">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M75">View MathML</a>

Now, we turn our attention to studying the derivative of the polynomials

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M76">View MathML</a>

(8)

By

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M77">View MathML</a>

and substituting <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M36">View MathML</a> in (8), we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M79">View MathML</a>

Hence, using the induction method, we arrive at the following result.

Theorem 2.6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M81">View MathML</a>

Now we give a generalization of the Raabe formula by the following theorem.

Theorem 2.7Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M82">View MathML</a>. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M83">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M84">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M85">View MathML</a>

Proof

We start the proof with defining the character

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M86">View MathML</a>

with the conductor <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M87">View MathML</a>. On the other hand, we derive that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M88">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M89">View MathML</a>

Then by using (6), it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M90">View MathML</a>

Now, we are ready to prove our result.

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M91">View MathML</a>

Hence, comparing the coefficient of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M46">View MathML</a> on both sides yields the assertion of this theorem. □

3 Decomposition of the generating function

In this section, using the generating element of the finite cyclic group G and the generating element of the subgroups of G, we show that the generating function with a conductor f can be written as a sum of the generating function with conductors which are less than f.

In this section, we use the following notations otherwise stated:

(i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M93">View MathML</a>,

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94">View MathML</a> denotes the integer such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M95">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M96">View MathML</a>,

(iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M97">View MathML</a>,

(iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M98">View MathML</a>,

(v) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M99">View MathML</a>.

We start to recall the fact that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M100">View MathML</a>

for positive integers x and y. In particular, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M101">View MathML</a>

whenever x and y are distinct prime numbers.

Theorem 3.1With the above notations, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M102">View MathML</a>

Proof We start to recall the fact that for sets A, B,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M103">View MathML</a>

Then by using De-Morgan’s law of sets, we get the following equality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M104">View MathML</a>

Now we use induction on n.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M105">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M106">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M107">View MathML</a>. We have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M108">View MathML</a>

We construct the following sets:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M109">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M110">View MathML</a>

By using De-Morgan’s law of sets, we verify the following equalities:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M111">View MathML</a>

and for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M112">View MathML</a>, it is clear that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M113">View MathML</a>.

Now assume that it is true for the set with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M114">View MathML</a> elements. Hence, the hypothesis for the sets <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M115">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M116">View MathML</a> is true, and we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M117">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M118">View MathML</a>

Hence it follows

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M119">View MathML</a>

 □

Example 3.2 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M120">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M121">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M122">View MathML</a> be distinct prime integers. Then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M123">View MathML</a>

Lemma 3.3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94">View MathML</a>be a prime number for alliand

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M125">View MathML</a>

Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M126">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M127">View MathML</a>is an empty set.

Proof By using the fact <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M128">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M129">View MathML</a> if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M130">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M131">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M42">View MathML</a>, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M133">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M134">View MathML</a>. Then we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M135">View MathML</a> for all i and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M136">View MathML</a>.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M137">View MathML</a>, then there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M131">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M139">View MathML</a> is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94">View MathML</a>. Therefore

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M141">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M142">View MathML</a>. Then there is <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M131">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M144">View MathML</a>. This means that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M145">View MathML</a> and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M137">View MathML</a>, a contradiction. Thus the proof is completed. □

By using the Lemma 3.3, we have one of the main results in this section.

Theorem 3.4With the above notations, we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M147">View MathML</a>

As stated before, to decompose the generating function of q-Eulerian polynomials attached to the character χ, now we need to compute the following relation:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M148">View MathML</a>

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M149">View MathML</a>

Also, we define the character <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M150">View MathML</a> with the conductor h such as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M151">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M152">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M153">View MathML</a>. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M154">View MathML</a>

Hence, we combine the above equation with the generating function

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M155">View MathML</a>

Now we are ready to state the main result without the proof in this section.

Theorem 3.5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M156">View MathML</a>, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94">View MathML</a>is a prime integer. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M158">View MathML</a>

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M12">View MathML</a>is a Dirichlet character corresponding to the characterχ.

Example 3.6 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M160">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M94">View MathML</a> is a prime integer. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/72/mathml/M162">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors completed the paper together. Both 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. The author would like to thank to all referees for their valuable comments and also for suggesting references [7-14].

References

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