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

  • Mustafa Alkan1Email author and

    Affiliated with

    • Yilmaz Simsek1

      Affiliated with

      Fixed Point Theory and Applications20132013:72

      DOI: 10.1186/1687-1812-2013-72

      Received: 28 December 2012

      Accepted: 6 March 2013

      Published: 27 March 2013

      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 [139]. 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 [24], 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-type Frobenius-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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq1_HTML.gif , the set of complex numbers, with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq2_HTML.gif
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equa_HTML.gif

      We use the following standard notions.

      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq4_HTML.gif and also, as usual, ℝ denotes the set of real numbers, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq5_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq6_HTML.gif (see [6]).

      A non-zero complex-valued function χ defined on G is called a character of G if for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq7_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equb_HTML.gif

      In particular, if G is a finite group with the identity element e, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq9_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq10_HTML.gif is defined as follows:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equc_HTML.gif
      Hence it is easily observed that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equd_HTML.gif

      for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq11_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq12_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq13_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq14_HTML.gif .

      (i) The q-Apostol type Frobenius-Euler numbers
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Eque_HTML.gif
      are defined by means of the following generating function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equf_HTML.gif
      (ii) The q-Apostol type Frobenius-Euler polynomials
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equg_HTML.gif
      are defined by means of the following generating function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equ1_HTML.gif
      (1)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equh_HTML.gif

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq12_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq13_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq15_HTML.gif ), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq17_HTML.gif ( http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq18_HTML.gif ). 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
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equi_HTML.gif
      are defined by means of the following generating function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equ2_HTML.gif
      (2)
      (ii) The q-Eulerian polynomials attached to the character χ
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equj_HTML.gif
      are defined by means of the following generating function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equ3_HTML.gif
      (3)
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equk_HTML.gif
      It is observed that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equl_HTML.gif
      Upon setting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq19_HTML.gif in (3), we compute a q-Eulerian number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq20_HTML.gif attached to the character χ as follows:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equm_HTML.gif
      By using the conductor f of the character χ and combining with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq21_HTML.gif , we modify Equation (2) and Equation (3), respectively, as follows:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equ4_HTML.gif
      (4)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equ5_HTML.gif
      (5)

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

      Theorem 2.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq22_HTML.gif . Then we have
      1. (i)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equn_HTML.gif
         
      2. (ii)
        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equo_HTML.gif
         
      Proof By using (4), we deduce that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equp_HTML.gif

      which, by comparing the coefficient http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq23_HTML.gif 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:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equq_HTML.gif

      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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq15_HTML.gif . The number http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq24_HTML.gif is defined by means of the following generating function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equr_HTML.gif
      The polynomials http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq25_HTML.gif are defined by means of the following generating function:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equ6_HTML.gif
      (6)
      Since we need this generating function frequently in this paper, we use the notation
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equs_HTML.gif

      and so it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq26_HTML.gif .

      By using the following well-known identity:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equt_HTML.gif
      in (3), we verify the following functional equation:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equ7_HTML.gif
      (7)

      Hence we have the following theorem.

      Theorem 2.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq22_HTML.gif . Then we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equu_HTML.gif
      Proof By applying the Cauchy product to (7), we deduce that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equv_HTML.gif
      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq27_HTML.gif . Then it follows that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equw_HTML.gif

      By comparing the coefficient of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq28_HTML.gif on both sides of the above equation, we obtain our desired result. □

      Upon setting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq29_HTML.gif in (7), we get the following functional equation:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equx_HTML.gif

      By substituting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq29_HTML.gif in Theorem 2.3, the following theorem is easily proved.

      Theorem 2.4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq22_HTML.gif . Then we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equy_HTML.gif
      So that we obtain a q-difference equation for q-Eulerian polynomials attached to the character χ, we study the following equations:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equz_HTML.gif
      which, in light of the Cauchy product of the three series http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq31_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equaa_HTML.gif

      yield the following theorem.

      Theorem 2.5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq22_HTML.gif . Then we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equab_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equac_HTML.gif
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equad_HTML.gif
      Now, we turn our attention to studying the derivative of the polynomials
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equ8_HTML.gif
      (8)
      By
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equae_HTML.gif
      and substituting http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq19_HTML.gif in (8), we get that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equaf_HTML.gif

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

      Theorem 2.6 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq22_HTML.gif . Then we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equag_HTML.gif

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

      Theorem 2.7 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq32_HTML.gif . Then we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equah_HTML.gif
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equai_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equaj_HTML.gif

      Proof

      We start the proof with defining the character
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equak_HTML.gif
      with the conductor http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq33_HTML.gif . On the other hand, we derive that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equal_HTML.gif
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equam_HTML.gif
      Then by using (6), it follows that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equan_HTML.gif
      Now, we are ready to prove our result.
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equao_HTML.gif

      Hence, comparing the coefficient of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq23_HTML.gif 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:
      1. (i)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq34_HTML.gif ,

         
      2. (ii)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq35_HTML.gif denotes the integer such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq36_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq37_HTML.gif ,

         
      3. (iii)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq38_HTML.gif ,

         
      4. (iv)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq39_HTML.gif ,

         
      5. (v)

        http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq40_HTML.gif .

         
      We start to recall the fact that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equap_HTML.gif
      for positive integers x and y. In particular, we get that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equaq_HTML.gif

      whenever x and y are distinct prime numbers.

      Theorem 3.1 With the above notations, we get
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equar_HTML.gif
      Proof We start to recall the fact that for sets A, B,
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equas_HTML.gif
      Then by using De-Morgan’s law of sets, we get the following equality:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equat_HTML.gif

      Now we use induction on n.

      If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq41_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq42_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq43_HTML.gif . We have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equau_HTML.gif
      We construct the following sets:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equav_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equaw_HTML.gif
      By using De-Morgan’s law of sets, we verify the following equalities:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equax_HTML.gif

      and for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq44_HTML.gif , it is clear that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq45_HTML.gif .

      Now assume that it is true for the set with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq46_HTML.gif elements. Hence, the hypothesis for the sets http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq47_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq48_HTML.gif is true, and we get that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equay_HTML.gif
      and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equaz_HTML.gif
      Hence it follows
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equba_HTML.gif

       □

      Example 3.2 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq49_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq50_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq51_HTML.gif be distinct prime integers. Then we have
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbb_HTML.gif
      Lemma 3.3 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq35_HTML.gif be a prime number for all i and
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbc_HTML.gif

      Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq52_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq53_HTML.gif is an empty set.

      Proof By using the fact http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq54_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq55_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq56_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq57_HTML.gif for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq22_HTML.gif , we get that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbd_HTML.gif

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq58_HTML.gif . Then we get that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq59_HTML.gif for all i and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq60_HTML.gif .

      If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq61_HTML.gif , then there is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq57_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq62_HTML.gif is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq35_HTML.gif . Therefore
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Eqube_HTML.gif

      Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq63_HTML.gif . Then there is http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq57_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq64_HTML.gif . This means that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq65_HTML.gif and so http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq61_HTML.gif , a contradiction. Thus the proof is completed. □

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

      Theorem 3.4 With the above notations, we get that
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbf_HTML.gif
      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:
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbg_HTML.gif
      where
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbh_HTML.gif
      Also, we define the character http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq66_HTML.gif with the conductor h such as
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbi_HTML.gif
      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq67_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq68_HTML.gif . Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbj_HTML.gif
      Hence, we combine the above equation with the generating function
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbk_HTML.gif

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

      Theorem 3.5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq69_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq35_HTML.gif is a prime integer. Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbl_HTML.gif

      where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq10_HTML.gif is a Dirichlet character corresponding to the character χ.

      Example 3.6 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq70_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_IEq35_HTML.gif is a prime integer. Then
      http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_409_Equbm_HTML.gif

      Declarations

      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 [714].

      Authors’ Affiliations

      (1)
      Department of Mathematics, Faculty of Science, Akdeniz University

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      Copyright

      © Alkan and Simsek; licensee Springer. 2013

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.