Open Access

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

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 q C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq1_HTML.gif, the set of complex numbers, with | q | < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq2_HTML.gif
[ x ] = [ x : q ] = { 1 q x 1 q if  q 1 , x if  q = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equa_HTML.gif

We use the following standard notions.

N = { 1 , 2 , } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq3_HTML.gif, N 0 = { 0 , 1 , 2 , } = N { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq4_HTML.gif and also, as usual, denotes the set of real numbers, R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq5_HTML.gif denotes the set of positive real numbers and 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 ( G , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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 a , b G https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq7_HTML.gif,
χ ( a b ) = χ ( a ) χ ( b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equb_HTML.gif

In particular, if G is a finite group with the identity element e, then χ ( e ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq8_HTML.gif and χ ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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 χ D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq10_HTML.gif is defined as follows:
χ D ( n ) = { χ ( n ) if  gcd ( n , f ) = 1 , 0 if  gcd ( n , f ) > 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equc_HTML.gif
Hence it is easily observed that
χ ( n f + a ) = χ ( a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equd_HTML.gif

for all n , a N https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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 a , b R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq12_HTML.gif ( a b https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq13_HTML.gif) and u C { 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq14_HTML.gif.
  1. (i)
    The q-Apostol type Frobenius-Euler numbers
    H n ( u ; a , b ; λ ; q ) ( λ , q C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Eque_HTML.gif
    are defined by means of the following generating function:
    F λ , q ( t ; u , a , b ) = ( 1 a t u ) n = 0 ( λ u ) n b [ n ] t = n = 0 H n ( u ; a , b ; λ ; q ) t n n ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equf_HTML.gif
     
  2. (ii)
    The q-Apostol type Frobenius-Euler polynomials
    H n ( x ; u ; a , b ; λ ; q ) ( λ C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equg_HTML.gif
    are defined by means of the following generating function:
    F λ , q ( x , t ; u , a , b ) = ( 1 a t u ) n = 0 ( λ u ) n b [ n + x ] t = n = 0 H n ( x ; u ; a , b ; λ ; q ) t n n ! , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equ1_HTML.gif
    (1)
    where
    | λ u b t | < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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 a , b R + https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq12_HTML.gif ( a b https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq13_HTML.gif and a 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq15_HTML.gif), u C { 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq16_HTML.gif, λ , q C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq17_HTML.gif ( | q | < 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq18_HTML.gif). Let χ be a character of a finite cyclic group G with the conductor f.
  1. (i)
    The q-Eulerian numbers attached to the character χ
    H n , χ ( u ; a , b ; λ ; q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equi_HTML.gif
    are defined by means of the following generating function:
    F λ , q , χ ( t , u , a , b ) = ( 1 a [ f ] t u f ) m = 0 ( λ u ) m b [ m ] t χ ( m ) = n = 0 H n , χ ( u ; a , b ; λ ; q ) t n n ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equ2_HTML.gif
    (2)
     
  2. (ii)
    The q-Eulerian polynomials attached to the character χ
    H n , χ ( x ; u ; a , b ; λ ; q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equj_HTML.gif
    are defined by means of the following generating function:
    F λ , q , χ ( t , x , u , a , b ) = ( 1 a [ f ] t u f ) m = 0 ( λ u ) m b [ m + x ] t χ ( m ) = n = 0 H n , χ ( x ; u ; a , b ; λ ; q ) t n n ! , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equ3_HTML.gif
    (3)
    where
    | λ u b t | < 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equk_HTML.gif
     
It is observed that
H n , χ ( 0 ; u ; a , b ; λ ; q ) = H n , χ ( u ; a , b ; λ ; q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equl_HTML.gif
Upon setting t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq19_HTML.gif in (3), we compute a q-Eulerian number H 0 , χ ( u , a , b , λ , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq20_HTML.gif attached to the character χ as follows:
H 0 , χ ( u , a , b , λ , q ) = ( u 1 f ( u f 1 ) u λ χ ( 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equm_HTML.gif
By using the conductor f of the character χ and combining with F λ , q ( t , x , u , a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq21_HTML.gif, we modify Equation (2) and Equation (3), respectively, as follows:
F λ , q , χ ( t , u , a , b ) = i = 0 f 1 ( λ χ ( 1 ) u ) i F λ f , q f ( t [ f ] , i f , u f , a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equ4_HTML.gif
(4)
and
F λ , q , χ ( t , x , u , a , b ) = i = 1 f ( λ χ ( 1 ) u ) i F λ f , q f ( [ f ] t , i + x f , u f , a , b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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 n N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq22_HTML.gif. Then we have
  1. (i)
    H n , χ ( u ; a , b ; λ , u ) = [ f ] n i = 0 f 1 ( λ u ) i χ ( i ) H n ( i f , u f , a , b ; λ f , q f ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equn_HTML.gif
     
  2. (ii)
    H n , χ ( u ; a , b ; λ , u ) = [ f ] n i = 0 f 1 χ ( i ) ( λ u ) i H n ( x + i f , u f , a , b ; λ f , q f ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equo_HTML.gif
     
Proof By using (4), we deduce that
n = 0 H n , χ ( u ; a , b ; λ , u ) t n n ! = i = 0 f 1 χ ( i ) ( λ u ) i n = 0 [ f ] n H n ( i f , u f , a , b ; λ f , q f ) t n n ! , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equp_HTML.gif

which, by comparing the coefficient t n n ! https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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:
H 0 , χ ( u , a , b , λ , q ) = i = 0 f 1 χ ( i ) ( λ u ) i H 0 ( i f , u f , a , b ; λ f , q f ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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 a 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq15_HTML.gif. The number Y n ( u , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq24_HTML.gif is defined by means of the following generating function:
G ( t , u , a ) = 1 a t u = n = 0 Y n ( u , a ) t n n ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equr_HTML.gif
The polynomials Y n ( x ; u , a ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq25_HTML.gif are defined by means of the following generating function:
G ( t , x , u , a ) = G ( t , u , a ) a x t = n = 0 Y n ( x ; u , a ) t n n ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equ6_HTML.gif
(6)
Since we need this generating function frequently in this paper, we use the notation
D ( k ) = a [ f ] t u f a [ f ] q k t u f = G ( [ f ] q k t , 1 q k , u f , a ) u f G ( [ f ] q k t , u f , a ) = n = 0 [ f ] n q k n ( Y n ( 1 q k , u f , a ) u f Y n ( u f , a ) ) t n n ! , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equs_HTML.gif

and so it follows that D n ( k ) = [ f ] n q n ( Y n ( 1 q , u f , a ) Y n ( u f , a ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq26_HTML.gif.

By using the following well-known identity:
[ n + x ] = [ x ] + q x [ n ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equt_HTML.gif
in (3), we verify the following functional equation:
F λ , q , χ ( t , x + y , u , a , b ) = b [ y ] t D ( y ) F λ , q , χ ( q y t , x , u , a , b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equ7_HTML.gif
(7)

Hence we have the following theorem.

Theorem 2.3 Let n N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq22_HTML.gif. Then we have
H n , χ ( x + y ; u ; a , b ; λ ; q ) = i = 0 m j = 0 i ( m i ) ( i j ) ( [ y ] ln b ) i j D j ( y ) q ( m i ) y × H ( m i ) i , χ ( x , u ; a , b ; λ ; q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equu_HTML.gif
Proof By applying the Cauchy product to (7), we deduce that
F λ , q , χ ( t , x + y , u , a , b ) = n = 0 j = 0 n ( n j ) ( [ y ] ln b ) n j D j ( y ) t n n ! F λ , q , χ ( t q y , x , u , a , b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equv_HTML.gif
Let K ( n ) = j = 0 n ( n j ) ( [ y ] ln b ) n j D j ( y ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq27_HTML.gif. Then it follows that
F λ , q , χ ( t , x + y , u , a , b ) = m = 0 i = 0 m ( m i ) K ( i ) q ( m i ) y H m i , χ ( x , u ; a , b ; λ ; q ) t m m ! = m = 0 i = 0 m j = 0 i ( m i ) ( i j ) ( [ y ] ln b ) i j D j ( y ) q ( m i ) y × H ( m i ) i , χ ( x , u ; a , b ; λ ; q ) t m m ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equw_HTML.gif

By comparing the coefficient of t m m ! https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq28_HTML.gif on both sides of the above equation, we obtain our desired result. □

Upon setting x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq29_HTML.gif in (7), we get the following functional equation:
F λ , q , χ ( t , y , u , a , b ) = b [ y ] t D ( y ) F λ , q , χ ( q y t , 0 , u , a , b ) = b [ y ] t D ( y ) F λ , q , χ ( q y t , u , a , b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equx_HTML.gif

By substituting x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq29_HTML.gif in Theorem 2.3, the following theorem is easily proved.

Theorem 2.4 Let n N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq22_HTML.gif. Then we have
H n , χ ( y ; u ; a , b ; λ ; q ) = i = 0 m j = 0 n i ( m i ) ( i j ) ( [ y ] ln b ) i j D j ( y ) q ( n i ) y H ( m i ) i , χ ( u ; a , b ; λ ; q ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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:
F λ , q , χ ( t , x , u , a , b ) = ( 1 a [ f ] t u f ) + λ χ ( 1 ) u F λ , q , χ ( t , x + 1 , u , a , b ) = ( 1 a [ f ] t u f ) + λ χ ( 1 ) b t u D ( 1 ) F λ , q , χ ( q t , x , u , a , b ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equz_HTML.gif
which, in light of the Cauchy product of the three series b t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq30_HTML.gif, F λ , q , χ ( q t , x , u , a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq31_HTML.gif and
D ( 1 ) = G ( q [ f ] t , 1 q , u f , a ) u f G ( q [ f ] t , u f , a ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equaa_HTML.gif

yield the following theorem.

Theorem 2.5 Let n N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq22_HTML.gif. Then we have
H 0 , χ ( x ; u ; a , b ; λ ; q ) = 1 ( [ f ] ln a ) n u f + λ χ ( 1 ) u C 0 H 0 , χ ( x ; u ; a , b ; λ ; q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equab_HTML.gif
and
H n , χ ( x ; u ; a , b ; λ ; q ) = ( [ f ] ln a ) n u f + λ χ ( 1 ) u i = 0 n ( n i ) C n i q i H i , χ ( x ; u ; a , b ; λ ; q ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equac_HTML.gif
where
C n = i = 0 n ( n i ) ( ln b ) n i [ f ] i q i ( Y i ( 1 q , u f , a ) u f Y i ( 0 , u , a ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equad_HTML.gif
Now, we turn our attention to studying the derivative of the polynomials
t F λ , q , χ ( x , t , u , a , b ) = [ f ] a [ f ] t ln a u f n = 0 ( λ u ) n χ ( n ) b [ n + x ] t + ( 1 a [ f ] t u f ) n = 0 ( λ u ) n χ ( n ) [ n + x ] b [ n + x ] t ln b . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equ8_HTML.gif
(8)
By
i = 0 y n [ n ] = 1 ( 1 y ) ( 1 q ) y ( 1 q y ) ( 1 q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equae_HTML.gif
and substituting t = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq19_HTML.gif in (8), we get that
H 1 , χ ( x ; u ; a , b ; λ ; q ) = [ f ] ln a u f u f 1 λ χ ( 1 ) + ( u f 1 u f ) ( u u λ χ ( 1 ) ) [ x ] ln b + ( u f 1 u f ( 1 q ) ) q x ( u u λ χ ( 1 ) + u u q λ χ ( 1 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equaf_HTML.gif

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

Theorem 2.6 Let n N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq22_HTML.gif. Then we have
H n , χ ( x ; u ; a , b ; λ ; q ) t n n ! = n t n F λ , q , χ ( x , t , u , a , b ) | t = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equag_HTML.gif

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

Theorem 2.7 Let c , n N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq32_HTML.gif. Then we have
H n , χ ( x ; u ; a , b ; λ ; q ) = n = 0 c 1 j = 0 n ( n j ) λ χ ( 1 ) u R j ( c ) [ c ] n j × H n j , χ c ( x + l c ; u c ; a , b ; λ c ; q c ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equah_HTML.gif
where
R j ( c ) = [ c f c ] j ( Y n ( [ f ] [ c f c ] ; u f , a ) u f Y n ( u f , a ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equai_HTML.gif
and
χ c ( x ) = χ ( c x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equaj_HTML.gif

Proof

We start the proof with defining the character
χ c ( x ) = χ ( c x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equak_HTML.gif
with the conductor f c = f gcd ( f , c ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq33_HTML.gif. On the other hand, we derive that
( u f a [ f ] t u f ) = u f a [ f ] t u f a [ f c ] q c [ c ] t u f a [ f c ] q c [ c ] t u f = R ( c ) u f a [ f c ] q c [ c ] t u f , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equal_HTML.gif
where
R ( c ) = a [ f ] t u f a [ c f c ] t u f . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equam_HTML.gif
Then by using (6), it follows that
R ( c ) = G ( [ c f c ] t , [ f ] [ c f c ] , u f , a ) u f G ( [ c f c ] t , 0 , u f , a ) = n = 0 [ c f c ] n ( Y n ( [ f ] [ c f c ] ; u f , a ) u f Y n ( u f , a ) ) t n n ! = n = 0 R n ( c ) t n n ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equan_HTML.gif
Now, we are ready to prove our result.
n = 0 H n , χ ( x ; u ; a , b ; λ ; q ) t n n ! = ( 1 a [ f ] t u f ) m = 0 l = 0 c 1 ( λ u ) l + m c b [ l + m c + x ] t χ ( l + m c ) = ( 1 a [ f ] t u f ) l = 0 c 1 ( λ χ ( 1 ) u ) l m = 0 ( λ c u c ) b [ m + l + x c ] q c [ c ] t χ ( c m ) = l = 0 c 1 ( λ χ ( 1 ) u ) l n = 0 R n ( c ) t n n ! n = 0 [ c ] n H n , χ c ( x + l c ; u c ; a , b ; λ c ; q c ) t n n ! = n = 0 l = 0 c 1 ( λ χ ( 1 ) u ) l j = 0 n ( n j ) R j ( c ) [ c ] n j H n j , χ c ( x + l c ; u c ; a , b ; λ c ; q c ) t n n ! . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equao_HTML.gif

Hence, comparing the coefficient of t n n ! https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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)

    x N 0 = { 0 , x , 2 x , 3 x , 4 x , } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq34_HTML.gif,

     
  2. (ii)

    p i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq35_HTML.gif denotes the integer such that gcd ( p i , p j ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq36_HTML.gif for all i , j { 1 , , n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq37_HTML.gif,

     
  3. (iii)

    S = i = 1 n ( p i N 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq38_HTML.gif,

     
  4. (iv)

    S 0 = { p 1 , , p n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq39_HTML.gif,

     
  5. (v)

    S i = { lcd ( a , b ) : a , b S i 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq40_HTML.gif.

     
We start to recall the fact that
x N 0 y N 0 = lcd ( x , y ) N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equap_HTML.gif
for positive integers x and y. In particular, we get that
x k N 0 y t N 0 = x y N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equaq_HTML.gif

whenever x and y are distinct prime numbers.

Theorem 3.1 With the above notations, we get
i S F ( i ) = j = 0 n 1 ( 1 ) j t S j i t N 0 F ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equar_HTML.gif
Proof We start to recall the fact that for sets A, B,
i A B F ( i ) = i A F ( i ) + i B F ( i ) i A B F ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equas_HTML.gif
Then by using De-Morgan’s law of sets, we get the following equality:
i ( p 1 N 0 ) ( i = 2 n ( p i N 0 ) ) F ( i ) = i ( p 1 N 0 ) F ( i ) + i ( i = 2 n ( p i N 0 ) ) F ( i ) i ( p 1 N 0 ) ( i = 2 n ( p i N 0 ) ) F ( i ) = i ( p 1 N 0 ) F ( i ) + i ( i = 2 n ( p i N 0 ) ) F ( i ) i ( i = 2 n lcd ( p 1 , p i ) N 0 ) F ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equat_HTML.gif

Now we use induction on n.

If n = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq41_HTML.gif, then S 0 = { p 1 , p 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq42_HTML.gif and S 1 = { lcd ( p 1 , p 2 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq43_HTML.gif. We have
i ( i = 1 2 ( p i N 0 ) ) F ( i ) = i ( p 1 N 0 ) F ( i ) + i ( p 2 N 0 ) F ( i ) i lcd ( p 1 , p 2 ) N 0 F ( i ) = j = 0 1 ( 1 ) j t S j i t N 0 F ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equau_HTML.gif
We construct the following sets:
S = i = 1 n ( p i N 0 ) , S = i = 2 n ( p i N 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equav_HTML.gif
and
S = ( p 1 N 0 ) S . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equaw_HTML.gif
By using De-Morgan’s law of sets, we verify the following equalities:
S = ( p 1 N 0 ) S = i = 2 n ( lcd ( p 1 , p i ) N 0 ) , S 0 = S 0 { p 1 } , S 1 = { lcd ( a , b ) : a , b S 0 } = S 1 S 0 , S 2 = { lcd ( a , b ) : a , b S 1 } = S 2 S 1 , S i = { lcd ( a , b ) : a , b S i 1 } = S i S i 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equax_HTML.gif

and for all i 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq44_HTML.gif, it is clear that S i S i 1 = https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq45_HTML.gif.

Now assume that it is true for the set with n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq46_HTML.gif elements. Hence, the hypothesis for the sets S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq47_HTML.gif and S https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq48_HTML.gif is true, and we get that
i S F ( i ) = j = 0 n 2 ( 1 ) j t S j i t N 0 F ( i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equay_HTML.gif
and
i S F ( i ) = j = 0 n 2 ( 1 ) j t S j i t N 0 F ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equaz_HTML.gif
Hence it follows
i S F ( i ) = i p 1 N 0 F ( i ) + i S F ( i ) i S F ( i ) = i p 1 N 0 F ( i ) + j = 0 n 2 ( 1 ) j t S j i t N 0 F ( i ) j = 0 n 2 ( 1 ) j t S j i t N 0 F ( i ) = t S 0 i t N 0 F ( i ) t S 1 S 0 i t N 0 F ( i ) + j = 2 n 2 ( 1 ) j t S j S j 1 i t N 0 F ( i ) = j = 0 n 1 ( 1 ) j t S j i t N 0 F ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equba_HTML.gif

 □

Example 3.2 Let p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq49_HTML.gif, p 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq50_HTML.gif, p 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq51_HTML.gif be distinct prime integers. Then we have
i ( i = 1 3 ( p i N 0 ) ) F ( i ) = i ( p 1 N 0 ) F ( i ) + i ( p 2 N 0 ) F ( i ) + i ( p 3 N 0 ) F ( i ) i ( p 2 p 3 N 0 ) F ( i ) i ( p 1 p 2 N 0 ) F ( i ) i ( p 1 p 3 N 0 ) F ( i ) + i ( p 1 p 2 p 3 N 0 ) F ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbb_HTML.gif
Lemma 3.3 Let p i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq35_HTML.gif be a prime number for all i and
C = { l N 0 : gcd ( f , l ) = 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbc_HTML.gif

Then N 0 = C ( i = 1 n ( p i N 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq52_HTML.gif and C ( i = 1 n ( p i N 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq53_HTML.gif is an empty set.

Proof By using the fact ( l , f ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq54_HTML.gif if and only if ( l , i = 1 n p i ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq55_HTML.gif if and only if ( l , p i ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq56_HTML.gif for all i { 1 , , n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq57_HTML.gif for n N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq22_HTML.gif, we get that
C = { l N 0 : gcd ( f , l ) = 1 } = { l N 0 :  for all  i { 1 , , n } , gcd ( p i , l ) = 1 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbd_HTML.gif

Let x i = 1 n ( p i N 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq58_HTML.gif. Then we get that gcd ( x , p i ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq59_HTML.gif for all i and so x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq60_HTML.gif.

If x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq61_HTML.gif, then there is i { 1 , , n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq57_HTML.gif such that gcd ( x , p i ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq62_HTML.gif is p i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq35_HTML.gif. Therefore
N 0 = C ( i = 1 n ( p i N 0 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Eqube_HTML.gif

Let x C ( i = 1 n ( p i N 0 ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq63_HTML.gif. Then there is i { 1 , , n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq57_HTML.gif such that x p i N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq64_HTML.gif. This means that gcd ( x , p i ) = p i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq65_HTML.gif and so x C https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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
i N 0 F ( i ) = i C F ( i ) + i ( j = 1 n ( p j N 0 ) ) F ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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:
n ( d Z ) ( λ u ) n b [ n ] t χ ( n ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbg_HTML.gif
where
f d = f d N 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbh_HTML.gif
Also, we define the character χ g https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq66_HTML.gif with the conductor h such as
χ g ( i ) : = χ ( g i ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbi_HTML.gif
where g , h N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq67_HTML.gif and gcd ( g , h ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq68_HTML.gif. Then
n ( d Z ) ( λ u ) n b [ n ] t χ ( n ) = n N 0 ( λ u ) d n b [ d n ] t χ ( d n ) = n N 0 i = 0 f d 1 ( λ u ) d ( f d n + i ) b [ d ( f d n + i ) ] t χ ( d ( f d n + i ) ) = n N 0 i = 0 f d 1 ( λ d u d ) f d n + i b t [ d ] [ f d n + i ] q d χ d ( i ) = i N 0 ( λ d u d ) i b t [ d ] [ i ] q d χ d ( i ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbj_HTML.gif
Hence, we combine the above equation with the generating function
( 1 a [ f ] t u f ) n ( d Z ) ( λ u ) n b [ n ] t χ ( n ) = ( 1 a [ d f d ] t u d f ) i N 0 ( λ u ) i b [ i ] q d t χ d ( i ) = ( 1 a [ d ] [ f d ] q d t ( u d ) f d ) i N 0 ( λ u ) i b d [ i ] q d t χ d ( i ) = F λ d , q d , χ d ( t , u d , a [ d ] , b [ d ] ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbk_HTML.gif

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

Theorem 3.5 Let f = i = 1 n p i t t https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq69_HTML.gif, where p i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq35_HTML.gif is a prime integer. Then
F λ , q , χ ( t , u , a , b ) = j = 0 n 1 ( 1 ) j t S j i t Z F λ i , q i , χ i ( t , u i , a [ i ] , b [ i ] ) + F λ , q , χ D ( t , u , a , b ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_Equbl_HTML.gif

where χ D https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq10_HTML.gif is a Dirichlet character corresponding to the character χ.

Example 3.6 Let f = p 1 t 1 p 2 t 2 p 3 t 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq70_HTML.gif, where p i https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_409_IEq35_HTML.gif is a prime integer. Then
F λ , q , χ ( t , u , a , b ) = F λ , q , χ D ( t , u , a , b ) + i = 1 3 F λ p i , q p i , χ p i ( t , u p i , a [ p i ] , b [ p i ] [ f p i ] q p i ) + F λ p 1 p 2 p 3 , q p 1 p 2 p 3 , χ p 1 p 2 p 3 ( t , u p 1 p 2 p 3 , a [ p 1 p 2 p 3 ] , b [ p 1 p 2 p 3 ] [ f p 1 p 2 p 3 ] q p 1 p 2 p 3 ) i = 1 3 F λ p 1 p i , q p 1 p i , χ p 1 p i ( t , u p 1 p i , a [ p 1 p i ] , b [ p 1 p i ] [ f p 1 p i ] q p 1 p i ) F λ p 2 p 3 , q p 2 p 3 , χ p 2 p 3 ( t , u p 2 p 3 , a [ p 2 p 3 ] , b [ p 2 p 3 ] [ f p 2 p 3 ] q p 2 p 3 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-72/MediaObjects/13663_2012_Article_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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