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 , the set of complex numbers, with | q | < 1
[ x ] = [ x : q ] = { 1 q x 1 q if  q 1 , x if  q = 1 .

We use the following standard notions.

N = { 1 , 2 , } , N 0 = { 0 , 1 , 2 , } = N { 0 } and also, as usual, denotes the set of real numbers, R + 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 , ) (see [6]).

A non-zero complex-valued function χ defined on G is called a character of G if for all a , b G ,
χ ( a b ) = χ ( a ) χ ( b ) .

In particular, if G is a finite group with the identity element e, then χ ( e ) = 1 and χ ( 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 χ D is defined as follows:
χ D ( n ) = { χ ( n ) if  gcd ( n , f ) = 1 , 0 if  gcd ( n , f ) > 1 .
Hence it is easily observed that
χ ( n f + a ) = χ ( a )

for all n , a N . 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 + ( a b ) and u C { 1 } .
  1. (i)
    The q-Apostol type Frobenius-Euler numbers
    H n ( u ; a , b ; λ ; q ) ( λ , q C )
    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 ! .
     
  2. (ii)
    The q-Apostol type Frobenius-Euler polynomials
    H n ( x ; u ; a , b ; λ ; q ) ( λ C )
    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 ! ,
    (1)
    where
    | λ u b t | < 1 .
     

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 + ( a b and a 1 ), u C { 1 } , λ , q C ( | q | < 1 ). 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 )
    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 ! .
    (2)
     
  2. (ii)
    The q-Eulerian polynomials attached to the character χ
    H n , χ ( x ; u ; a , b ; λ ; q )
    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 ! ,
    (3)
    where
    | λ u b t | < 1 .
     
It is observed that
H n , χ ( 0 ; u ; a , b ; λ ; q ) = H n , χ ( u ; a , b ; λ ; q ) .
Upon setting t = 0 in (3), we compute a q-Eulerian number H 0 , χ ( u , a , b , λ , q ) 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 ) ) .
By using the conductor f of the character χ and combining with F λ , q ( t , x , u , a , b ) , 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 )
(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 ) .
(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 . 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 ) ,
     
  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 ) .
     
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 ! ,

which, by comparing the coefficient t n n ! 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 ) .

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 . The number Y n ( u , a ) 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 ! .
The polynomials Y n ( x ; u , a ) 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 ! .
(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 ! ,

and so it follows that D n ( k ) = [ f ] n q n ( Y n ( 1 q , u f , a ) Y n ( u f , a ) ) .

By using the following well-known identity:
[ n + x ] = [ x ] + q x [ n ]
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 ) .
(7)

Hence we have the following theorem.

Theorem 2.3 Let n N 0 . 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 ) .
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 ) .
Let K ( n ) = j = 0 n ( n j ) ( [ y ] ln b ) n j D j ( y ) . 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 ! .

By comparing the coefficient of t m m ! on both sides of the above equation, we obtain our desired result. □

Upon setting x = 0 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 ) .

By substituting x = 0 in Theorem 2.3, the following theorem is easily proved.

Theorem 2.4 Let n N 0 . 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 ) .
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 ) ,
which, in light of the Cauchy product of the three series b t , F λ , q , χ ( q t , x , u , a , b ) and
D ( 1 ) = G ( q [ f ] t , 1 q , u f , a ) u f G ( q [ f ] t , u f , a ) ,

yield the following theorem.

Theorem 2.5 Let n N 0 . 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 )
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 ) ,
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 ) ) .
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 .
(8)
By
i = 0 y n [ n ] = 1 ( 1 y ) ( 1 q ) y ( 1 q y ) ( 1 q )
and substituting t = 0 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 ) ) .

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

Theorem 2.6 Let n N 0 . 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 .

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

Theorem 2.7 Let c , n N 0 . 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 ) ,
where
R j ( c ) = [ c f c ] j ( Y n ( [ f ] [ c f c ] ; u f , a ) u f Y n ( u f , a ) )
and
χ c ( x ) = χ ( c x ) .

Proof

We start the proof with defining the character
χ c ( x ) = χ ( c x )
with the conductor f c = f gcd ( f , c ) . 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 ,
where
R ( c ) = a [ f ] t u f a [ c f c ] t u f .
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 ! .
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 ! .

Hence, comparing the coefficient of t n n ! 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 , } ,

     
  2. (ii)

    p i denotes the integer such that gcd ( p i , p j ) = 1 for all i , j { 1 , , n } ,

     
  3. (iii)

    S = i = 1 n ( p i N 0 ) ,

     
  4. (iv)

    S 0 = { p 1 , , p n } ,

     
  5. (v)

    S i = { lcd ( a , b ) : a , b S i 1 } .

     
We start to recall the fact that
x N 0 y N 0 = lcd ( x , y ) N 0
for positive integers x and y. In particular, we get that
x k N 0 y t N 0 = x y N 0

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 ) .
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 ) .
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 ) .

Now we use induction on n.

If n = 2 , then S 0 = { p 1 , p 2 } and S 1 = { lcd ( p 1 , p 2 ) } . 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 ) .
We construct the following sets:
S = i = 1 n ( p i N 0 ) , S = i = 2 n ( p i N 0 )
and
S = ( p 1 N 0 ) S .
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

and for all i 1 , it is clear that S i S i 1 = .

Now assume that it is true for the set with n 1 elements. Hence, the hypothesis for the sets S and S 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 )
and
i S F ( i ) = j = 0 n 2 ( 1 ) j t S j i t N 0 F ( i ) .
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 ) .

 □

Example 3.2 Let p 1 , p 2 , p 3 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 ) .
Lemma 3.3 Let p i be a prime number for all i and
C = { l N 0 : gcd ( f , l ) = 1 } .

Then N 0 = C ( i = 1 n ( p i N 0 ) ) and C ( i = 1 n ( p i N 0 ) ) is an empty set.

Proof By using the fact ( l , f ) = 1 if and only if ( l , i = 1 n p i ) = 1 if and only if ( l , p i ) = 1 for all i { 1 , , n } for n N 0 , we get that
C = { l N 0 : gcd ( f , l ) = 1 } = { l N 0 :  for all  i { 1 , , n } , gcd ( p i , l ) = 1 } .

Let x i = 1 n ( p i N 0 ) . Then we get that gcd ( x , p i ) = 1 for all i and so x C .

If x C , then there is i { 1 , , n } such that gcd ( x , p i ) is p i . Therefore
N 0 = C ( i = 1 n ( p i N 0 ) ) .

Let x C ( i = 1 n ( p i N 0 ) ) . Then there is i { 1 , , n } such that x p i N 0 . This means that gcd ( x , p i ) = p i and so x C , 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 ) .
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 ) ,
where
f d = f d N 0 .
Also, we define the character χ g with the conductor h such as
χ g ( i ) : = χ ( g i ) ,
where g , h N 0 and gcd ( g , h ) = 1 . 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 ) .
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 ] ) .

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 , where p i 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 ) ,

where χ D 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 , where p i 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 ) .

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

References

  1. Simsek Y: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications. Fixed Point Theory Appl. 2013., 2013: Article ID 87. doi:10.1186/1687–1812–2013–87
  2. Cangül IN, Cevik AS, Simsek Y: Analysis approach to finite monoids. Fixed Point Theory Appl. 2013., 2013: Article ID 15. doi:10.1186/1687–1812–2013–15
  3. Cevik AS, Cangül IN, Simsek Y: A new approach to connect algebra with analysis: relationship and applications between presentations and generating functions. Bound. Value Probl. 2013., 2013: Article ID 51. doi:10.1186/1687–2770–2013–51
  4. Cevik AS, Das KC, Simsek Y, Cangül IN: Some array polynomials over special monoids presentations. Fixed Point Theory Appl. 2013., 2013: Article ID 44. doi:10.1186/1687–1812–2013–44
  5. Simsek Y: Generating functions for q -Apostol type Frobenius-Euler numbers and polynomials. Axioms 2012, 1: 395–403. doi:10.3390/axioms1030395 10.3390/axioms1030395View Article
  6. Apostol TM: Introduction to Analytic Number Theory. Springer, New York; 1976.
  7. Luo QM, Srivastava HM: Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 2005, 10: 631–642.MathSciNet
  8. Luo QM, Srivastava HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 2005, 308: 290–302. 10.1016/j.jmaa.2005.01.020MathSciNetView Article
  9. Luo QM: Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwan. J. Math. 2006, 10(4):917–925.
  10. Luo QM, Srivastava HM: Some relationships between Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 2006, 51: 631–642. 10.1016/j.camwa.2005.04.018MathSciNetView Article
  11. Luo QM, Srivastava HM: q -extensions of some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Taiwan. J. Math. 2011, 15: 241–257.MathSciNet
  12. Luo QM: Some results for the q -Bernoulli and q -Euler polynomials. J. Math. Anal. Appl. 2010, 363: 7–18. 10.1016/j.jmaa.2009.07.042MathSciNetView Article
  13. Luo QM: An explicit relationships between the generalized Apostol-Bernoulli and Apostol-Euler polynomials associated with λ -Stirling numbers of the second kind. Houst. J. Math. 2010, 36: 1159–1171.
  14. Luo QM: Some formulas for the Apostol-Euler polynomials associated with Hurwitz zeta function at rational arguments. Appl. Anal. Discrete Math. 2009, 3: 336–346. 10.2298/AADM0902336LMathSciNetView Article
  15. Choi J, Anderson PJ, Srivastava HM: Carlitz’s q -Bernoulli and q -Euler numbers and polynomials and a class of generalized q -Hurwitz zeta functions. Appl. Math. Comput. 2009, 215: 1185–1208. 10.1016/j.amc.2009.06.060MathSciNetView Article
  16. Kim T, Jang CL, Park HK: A note on q -Euler and Genocchi numbers. Proc. Jpn. Acad. 2001, 77: 139–141. 10.3792/pjaa.77.139View Article
  17. Simsek Y, Bayad A, Lokesha V: q -Bernstein polynomials related to q -Frobenius-Euler polynomials, l -functions, and q -Stirling numbers. Math. Methods Appl. Sci. 2012, 35: 877–884. 10.1002/mma.1580MathSciNetView Article
  18. Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.
  19. Srivastava HM: Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 2011, 5: 390–444.MathSciNet
  20. Kim T: A note on some formulae for the q -Euler numbers and polynomials. Proc. Jangjeon Math. Soc. 2006, 9: 227–232. arXiv:math/0608649MathSciNet
  21. Kim T: The modified q -Euler numbers and polynomials. Adv. Stud. Contemp. Math. 2008, 16: 161–170. arXiv:math/0702523
  22. Kim T: A note on the alternating sums of powers of consecutive q -integers. Adv. Stud. Contemp. Math. 2005, 11: 137–140. arXiv:math/0604227v1 [math.NT]
  23. Carlitz L: q -Bernoulli and Eulerian numbers. Trans. Am. Math. Soc. 1954, 76: 332–350.
  24. Choi J, Anderson PJ, Srivastava HM: Some q -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n , and the multiple Hurwitz zeta function. Appl. Math. Comput. 2008, 199: 723–737. 10.1016/j.amc.2007.10.033MathSciNetView Article
  25. Kim T: On a q -analogue of the p -adic log gamma functions and related integrals. J. Number Theory 1999, 76: 320–329. 10.1006/jnth.1999.2373MathSciNetView Article
  26. Jang LC, Kim T: q -analogue of Euler-Barnes’ numbers and polynomials. Bull. Korean Math. Soc. 2005, 42: 491–499.MathSciNetView Article
  27. Kim T: On the q -extension of Euler and Genocchi numbers. J. Math. Anal. Appl. 2007, 326: 1458–1465. 10.1016/j.jmaa.2006.03.037MathSciNetView Article
  28. Kim T, Jang LC, Rim SH, Pak HK: On the twisted q -zeta functions and q -Bernoulli polynomials. Far East J. Appl. Math. 2003, 13: 13–21.MathSciNet
  29. Kim T, Rim SH: A new Changhee q -Euler numbers and polynomials associated with p -adic q -integral. Comput. Math. Appl. 2007, 54: 484–489. 10.1016/j.camwa.2006.12.028MathSciNetView Article
  30. Ozden H, Simsek Y: A new extension of q -Euler numbers and polynomials related to their interpolation functions. Appl. Math. Lett. 2008, 21: 934–939. 10.1016/j.aml.2007.10.005MathSciNetView Article
  31. Ozden H, Simsek Y, Srivastava HM: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 2010, 60: 2779–2787. 10.1016/j.camwa.2010.09.031MathSciNetView Article
  32. Schempp W: Euler-Frobenius polynomials. Internat. Schriftenreihe Numer. Math. 67. Numer. Methods Approx. Theory, vol. 7 (Oberwolfach, 1983) 1984, 131–138.
  33. Shiratani K: On Euler numbers. Mem. Fac. Sci. Kyushu Univ. 1975, 27: 1–5.MathSciNet
  34. Simsek Y: Twisted ( h , q ) -Bernoulli numbers and polynomials related to twisted ( h , q ) -zeta function and L -function. J. Math. Anal. Appl. 2006, 324: 790–804. 10.1016/j.jmaa.2005.12.057MathSciNetView Article
  35. Simsek Y: q -analogue of the twisted l -series and q -twisted Euler numbers. J. Number Theory 2005, 110: 267–278. 10.1016/j.jnt.2004.07.003MathSciNetView Article
  36. Simsek Y: On p -adic twisted q - L -functions related to generalized twisted Bernoulli numbers. Russ. J. Math. Phys. 2006, 13: 327–339.MathSciNetView Article
  37. Simsek Y, Kim T, Park DW, Ro YS, Jang LC, Rim SH: An explicit formula for the multiple Frobenius-Euler numbers and polynomials. JP J. Algebra Number Theory Appl. 2004, 4: 519–529.MathSciNet
  38. Srivastava HM, Kim T, Simsek Y: q -Bernoulli numbers and polynomials associated with multiple q -zeta functions and basic L -series. Russ. J. Math. Phys. 2005, 12: 241–268.MathSciNet
  39. Tsumura H: A note on q -analogues of the Dirichlet series and q -Bernoulli numbers. J. Number Theory 1991, 39: 251–256. 10.1016/0022-314X(91)90048-GMathSciNetView Article

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.