Open Access

Some identities on the higher-order-twisted q-Euler numbers and polynomials with weight α

  • Hui Young Lee1Email author,
  • Nam Soon Jung1,
  • Jung Yoog Kang1 and
  • Cheon Seoung Ryoo1
Advances in Difference Equations20122012:21

https://doi.org/10.1186/1687-1847-2012-21

Received: 19 September 2011

Accepted: 29 February 2012

Published: 29 February 2012

Abstract

In this article, we introduce some properties of higher-order-twisted q-Euler numbers and polynomials with weight α, and we observe some properties of higher-order-twisted q-Euler numbers and polynomials with weight α for several cases. In particular, by using the the fermionic p-adic q-integral on p , we give a new concept of twisted q-Euler numbers and polynomials with weight α.

2000 Mathematics Subject Classification: 11B68; 11S40; 11S80.

Keywords

Euler numbers and polynomials q-Euler numbers and polynomials higher-order-twisted q-Euler numbers and polynomials with weight α

1. Introduction

Let p be a fixed odd prime. Throughout this article p , p , , and p , will, respectively, denote the ring of p- adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of p . Let be the set of natural numbers and + = {0}. Let ν p be the normalized exponential valuation of p with p p = p - ν p ( p ) = p - 1 (see [114]). When one speaks of q-extension, q can be regarded as an indeterminate, a complex number q , or p-adic number q p ; it is always clear from context. If q , we assume |q| < 1. If q p , then we assume |1 - q| p < 1 (see [114]).

In this article, we use the notation of q-number as follows (see [114]):
[ x ] q = 1 - q x 1 - q .

Note that limq→1[x] q = x for any x with |x| p ≤ 1 in the p-adic case.

Let C( p ) be the space of continuous functions on p . For f C( p ), Kim defined the fermionic p-adic q-integral on p as follows (see [6, 7]):
I - q ( f ) = p f ( x ) d μ - q ( x ) = lim N 1 [ p N ] - q x = 0 p N - 1 f ( x ) ( - q ) x , = lim N [ 2 ] q 1 + q p N x = 0 p N - 1 f ( x ) ( - q ) x .
(1)
From (1), we note that
q I - q ( f 1 ) + I - q ( f ) = [ 2 ] q f ( 0 ) ,

where f1(x) = f(x + 1).

It is well known that the ordinary Euler polynomials are defined by
2 e t + 1 e x t = e E ( x ) t = n = 0 E n ( x ) t n n ! ,

with the usual convention of replacing E n (x) by E n (x).

In the special case, x = 0, E n (0) = E n are called the n th Euler numbers (see [114]).

By (2), we get the following recurrence relation as follows:
E 0 = 1 , and ( E + 1 ) n + E = 2 , if n = 0 , 0 , if n > 0 .
(2)
Recently, (h, q)-Euler numbers are defined by
E 0 , q ( h ) = 2 1 + q h , and q h ( q E q ( h ) + 1 ) n + E q ( h ) = 2 , if n = 0 , 0 , if n > 0 ,

with the usual convention about replacing E q ( h ) n by E n , q ( h ) (see [116]).

Note that lim q 1 E n , q ( h ) = E n .

Let T p = N 1 C p N = lim N C p N , where C p N = { w | w p N = 1 } is the cyclic group of order p N . For w T p , we denote by ϕ w : p p the locally constant function x w x .

For α and w T p , the twisted q-Euler numbers with weight α are also defined by
0 , q , w ( α ) = [ 2 ] q w q + 1 , and w q q α q , w ( α ) + 1 n + n , q , w ( α ) = [ 2 ] q , if n = 0 , 0 , if n > 0 ,

with the usual convention about replacing q , w ( α ) n by n , q , w ( α ) (see [2, 5]).

The main purpose of this article is to present a systemic study of some families of higher-order-twisted q-Euler numbers and polynomials with weight α. In Section 2, we investigate higher-order-twisted q-Euler numbers and polynomials with weight α and establish interesting properties. In Sections 3, 4, and 5, we observe some properties for special cases.

2. Higher-order-twisted q-Euler numbers and polynomials with weight α

For h , α, k , w T p and n +, let us consider the expansion of higher-order-twisted q-Euler polynomials with weight α as follows:
n , q , w ( α ) ( h , k | x ) = p p w k - times i = 1 k x i i = 1 k x i + x q α n q x 1 ( h - 1 ) + + x k ( h - k ) d μ - q ( x 1 ) d μ - q ( x k ) .
(3)
From (1) and (3), we note that
n , q , w ( α ) ( h , k | x ) = [ 2 ] q k ( 1 - q α ) n l = 0 n n l ( - 1 ) l q α l x ( 1 + w q α l + h ) ( 1 + w q α l + h - k + 1 ) .
(4)

In the special case, x = 0 n , q , w ( α ) ( h , k | 0 ) = n , q , w ( α ) ( h , k ) are called the higher-order-twisted q-Euler numbers with weight α.

By (3), we get
n , q , w ( α ) ( h , k ) = ( q α - 1 ) n + 1 , q , w ( α ) ( h - α , k ) + n , q , w ( α ) ( h - α , k ) .
(5)

From (5) and mathematical induction, we get the following theorem.

Theorem 1. For α, k and n +, we have
i = 0 n - 1 ( - 1 ) i - 1 ( q α - 1 ) n - 1 - i n - i , q , w ( α ) ( h , k ) = ( q α - 1 ) n - 1 n , q , w ( α ) ( h - α , k ) + ( - 1 ) n E 1 , q , w ( α ) ( h - α , k ) .
For complex number q p , m +, we get the following;
q α ( x 1 + + x k + 1 ) m = 1 - ( 1 - q α ( x 1 + + x k + 1 ) ) m = l = 0 m m l ( - 1 ) l 1 - q α ( x 1 + + x k + 1 ) l = l = 0 m m l ( - 1 ) l ( 1 - q α ) l ( 1 - q α ( x 1 + + x k + 1 ) ) l ( 1 - q α ) l = l = 0 m m l ( - 1 ) l ( 1 - q α ) l [ x 1 + x 2 + + x k + 1 ] q α l .
From (3), (4), and above property, we have
0 , q , w ( α ) ( m α , k + 1 ) = p p w j = 1 k + 1 x j q j = 1 k + 1 ( m α - j ) x j d μ - q ( x 1 ) d μ - q ( x k + 1 ) = l = 0 m m l ( q α - 1 ) l p p w j = 1 k + 1 x j j = 1 k + 1 x j q α l q - j = 1 k + 1 j x j d μ - q ( x 1 ) d μ - q ( x k + 1 ) = l = 0 m m l ( q α - 1 ) l l , q , w ( α ) ( 0 , k + 1 ) = [ 2 ] q k + 1 ( 1 + w q α m ) ( 1 + w q α m - 1 ) ( 1 + w q α m - k ) .
(6)
From (3), we can derive the following equation.
j = 0 i i j ( q α - 1 ) j p p w s = 1 k x s s = 1 k x s q α n - i + j q s = 1 k ( h - α - s ) x s d μ - q ( x 1 ) d μ - q ( x k ) = p p w s = 1 k x s s = 1 k x s q α n - i q s = 1 k ( h - s ) x s q α s = 1 k x s ( i - 1 ) d μ - q ( x 1 ) d μ - q ( x k ) = j = 0 i - 1 ( q α - 1 ) j i - 1 j n - i + j , q , w ( α ) ( h , k ) .
(7)
By (3), (4), (5), and (6), we see that
j = 0 i ( q α - 1 ) j i j n - 1 + j , q , w ( α ) ( h - α , k ) = j = 0 i - 1 ( q α - 1 ) j i - 1 j n - i + j , q , w ( α ) ( h , k ) .

Therefore, we obtain the following theorem.

Theorem 2. For α, k and n, i +, we have
j = 0 i i j ( q α - 1 ) j n - i + j , q , w ( α ) ( h - α , k ) = j = 0 i - 1 ( q α - 1 ) j i - 1 j n - i + j , q , w ( α ) ( h , k ) .
By simple calculation, we easily see that
j = 0 m m j ( q α - 1 ) j j , q , w ( α ) ( 0 , k ) = [ 2 ] q k ( 1 + w q α m ) ( 1 + w q α m - 1 ) ( 1 + w q α m - k + 1 ) .

3. Polynomials n , q , w ( α ) ( 0 , k | x )

We now consider the polynomials n , q , w ( α ) ( 0 , k | x ) (in q x ) by
n , q , w ( α ) ( 0 , k | x ) = p p k - times w x 1 + + x k x + i = 1 k x i q α n q - j = 1 k j x j d μ - q ( x 1 ) d μ - q ( x k ) .
(8)
By (8) and (4), we get
( q α - 1 ) n n , q , w ( α ) ( 0 , k | x ) = [ 2 ] q k l = 0 n n l q α l x ( - 1 ) n - 1 1 ( 1 + w q α l ) ( 1 + w q α l - k + 1 ) .
(9)
From (8) and (9), we can derive the following equation.
p p w x 1 + + x k q j = 1 k ( α n - j ) x j + α n x d μ - q ( x 1 ) d μ - q ( x k ) = j = 0 n n j [ α ] q j ( q - 1 ) j j , q , w ( α ) ( 0 , k | x ) ,
and
p p w x 1 + + x k q j = 1 k ( α n - j ) x j + α n x d μ - q ( x 1 ) d μ - q ( x k ) = [ 2 ] q k q α n x ( 1 + w q α n ) ( 1 + w q α n - k + 1 ) .
(10)

Therefore, by (9) and (10), we obtain the following theorem.

Theorem 3. For α and n, k +, we have
n , q , w ( α ) ( 0 , k | x ) = [ 2 ] q k [ α ] q n ( 1 - q ) n l = 0 n n l ( - 1 ) l q α l x 1 ( - w q α l - k + 1 : q ) k ,
and
l = 0 n n l [ α ] q l ( q - 1 ) l l , q , w ( α ) ( 0 , k | x ) = q α n x [ 2 ] q k ( - w q α n - k + 1 : q ) k ,

where (a : q)0 = 1 and (a : q) k = (1 - a)(1 - aq) (1 - aqk-1).

Let d with d ≡ 1 (mod 2). Then we have
p p w x 1 + + x k x + j = 1 k x j q α n q - j = 1 k j x j d μ - q ( x 1 ) d μ - q ( x k ) = [ d ] q α n [ d ] - q k a 1 , , a k = 0 d - 1 w a 1 + + a k q - j = 2 k ( j - 1 ) a j ( - 1 ) j = 1 k a j × p p w d ( x 1 + + x k ) x + j = 1 k a j d + j = 1 k x j q α d n q - d j = 1 k j x j d μ - q d ( x 1 ) d μ - q d ( x k )
(11)

Thus, by (11), we obtain the following theorem.

Theorem 4. For d with d ≡ 1 (mod 2), we have
n , q , w ( α ) ( 0 , k | x ) = [ d ] q α n [ d ] - q k a 1 , , a k = 0 d - 1 ( - w ) a 1 + + a k q - j = 2 k ( j - 1 ) a j n , q d , w d ( α ) 0 , k | x + a 1 + + a k d .
Moreover,
n , q , w ( α ) ( 0 , k | d x ) = [ d ] q α n [ d ] - q k a 1 , , a k = 0 d - 1 ( - w ) a 1 + + a k q - j = 2 k ( j - 1 ) a j n , q d , w d ( α ) 0 , k | x + a 1 + + a k d .
By (8), we get
E ˜ n , q , w ( α ) ( 0 , k | x = l = 0 n ( n l ) [ x ] q α n l q α l x E ˜ l , q , w ( α ) ( 0 , k ) = ( [ x ] q α + q α x E ˜ q , w ( α ) ( 0 , k ) ) n ,

where n , q , w ( α ) ( 0 , k | 0 ) = n , q , w ( α ) ( 0 , k ) .

Thus, we note that
E ˜ n , q , w ( α ) ( 0 , k | x + y ) = l = 0 n ( n l ) [ y ] q α n l q α l y E ˜ l , q , w ( α ) ( 0 , k | x ) = ( [ y ] q α + q α y E ˜ q , w ( α ) ( 0 , k | x ) ) n .

4. Polynomials n , q , w ( α ) ( h , 1 | x )

Let us define polynomials n , q , w ( α ) ( h , 1 | x ) as follows:
n , q , w ( α ) ( h , 1 | x ) = p w x 1 [ x + x 1 ] q α n q x 1 ( h - 1 ) d μ - q ( x 1 ) .
(12)
From (12), we have
n , q , w ( α ) ( h , 1 | x ) = [ 2 ] q ( 1 - q α ) n l = 0 n n l ( - 1 ) l q α l x 1 ( 1 + w q α l + h ) .
By the calculation of the fermionic p-adic q- integral on p , we see that
q α x p w x 1 [ x + x 1 ] q α n q x 1 ( h - 1 ) d μ - q ( x 1 ) = ( q α - 1 ) p w x 1 [ x + x 1 ] q α n + 1 q x 1 ( h - α - 1 ) d μ - q ( x 1 ) + p w x 1 [ x + x 1 ] q α n q x 1 ( h - α - 1 ) d μ - q ( x 1 ) .
(13)

Thus, by (13), we obtain the following theorem.

Theorem 5. For α and h , we have
q α x n , q , w ( α ) ( h , 1 | x ) = ( q α - 1 ) n + 1 , q , w ( α ) ( h - α , 1 | x ) + n , q , w ( α ) ( h - α , 1 | x ) .
It is easy to show that
n , q , w ( α ) ( h , 1 | x ) = p w x 1 [ x + x 1 ] q α n q x 1 ( h - 1 ) d μ - q ( x 1 ) = l = 0 n n l [ x ] q α n - 1 q α l x p w x 1 [ x 1 ] q α l q x 1 ( h - 1 ) d μ - q ( x 1 ) = l = 0 n n l [ x ] q α n - 1 q α l x l , q , w ( α ) ( h , 1 ) = q α x q , w ( α ) ( h , 1 ) + [ x ] q α n , for n 1 ,
(14)

with the usual convention about replacing ( q , w ( α ) ( h , 1 ) ) n by n , q , w ( α ) ( h , 1 ) .

From qI-q(f1) + I-q(f) = [2] q f(0), we have
w q h p w x 1 [ x + x 1 + 1 ] q α n q x 1 ( h - 1 ) d μ - q ( x 1 ) + p w x 1 [ x + x 1 ] q α n q x 1 ( h - 1 ) d μ - q ( x 1 ) = [ 2 ] q [ x ] q α n .
(15)
By (13) and (15), we get
w q h n , q , w ( α ) ( h , 1 | x + 1 ) + n , q , w ( α ) ( h , 1 | x ) = [ 2 ] q [ x ] q α n .
(16)
For x = 0 in (16), we have
w q h n , q , w ( α ) ( h , 1 | 1 ) + n , q , w ( α ) ( h , 1 ) = [ 2 ] q , if n = 0 , 0 , if n > 0 .
(17)

Therefore, by (14) and (17), we obtain the following theorem.

Theorem 6. For h and n +, we have
w q h ( q α q , w ( α ) ( h , 1 ) + 1 ) n n , q , w ( α ) ( h , 1 ) = [ 2 ] q , if n = 0 , 0 , if n > 0 ,

with the usual convention about replacing ( q , w ( α ) ( h , 1 ) ) n by n , q , w ( α ) ( h , 1 ) .

From the fermionic p-adic q-integral on p , we easily get
0 , q , w ( α ) ( h , 1 ) = p w x 1 q x 1 ( h - 1 ) d μ - q ( x 1 ) = [ 2 ] q [ 2 ] w q h .
By (12), we see that
n , q - 1 , w - 1 ( α ) ( h , 1 | 1 - x ) = p w - 1 [ 1 - x + x 1 ] q - α n q - x 1 ( h - 1 ) d μ - q - 1 ( x 1 ) = ( - 1 ) n w q α n + h - 1 [ 2 ] q ( 1 - q α ) n l = 0 n n l ( - 1 ) l q α l x 1 1 + w q α l + h = ( - 1 ) n w q α n + h - 1 n , q , w ( α ) ( h , 1 | x )
(18)

Therefore, by (18), we obtain the following theorem.

Theorem 7. For α , h and n +, we have
n , q - 1 , w - 1 ( α ) ( h , 1 | 1 - x ) = ( - 1 ) n w q α n + h - 1 n , q , w ( α ) ( h , 1 | x ) .
In particular, for x = 1, we get
n , q , w ( α ) ( h , 1 ) = ( - 1 ) n w q α n + h - 1 n , q , w ( α ) ( h , 1 | 1 ) = ( - 1 ) n + 1 q α n - 1 n , q , w ( α ) ( h , 1 ) if n 1 .
Let d with d ≡ 1 (mod 2). Then we have
p w x 1 q x 1 ( h - 1 ) [ x + x 1 ] q α n d μ - q ( x 1 ) = [ d ] q α n [ d ] - q a = 0 d - 1 w a q h a ( - 1 ) a p w d x 1 x + a d + x 1 q α d n q x 1 ( h - 1 ) d d μ - q d ( x 1 ) .
(19)

Therefore, by (19), we obtain the following theorem.

Theorem 8 (Multiplication formula). For d with d ≡ 1 (mod 2), we have
n , q , w ( α ) ( h , 1 | x ) = [ d ] q α n [ d ] - q a = 0 d - 1 w a q h a ( - 1 ) q n , q d , w d ( α ) h , 1 | x + a d .

5. Polynomials n , q , w ( α ) ( h , k | x ) and k= h

In (3), we know that
n , q , w ( α ) ( h , k | x ) = p p w x 1 + + x k [ x 1 + + x k + x ] q α n q ( h - 1 ) x 1 + + ( h - k ) x k d μ - q ( x 1 ) d μ - q ( x k ) .
Thus, we get
( q α - 1 ) n n , q , w ( α ) ( h , k | x ) = [ 2 ] q k l = 0 n n l ( - 1 ) n - l q α l x ( 1 + w q α l + h ) ( 1 + w q α l + h - k + 1 ) ,
and
w q h p p w x 1 + + x k [ x + 1 + i = 1 k x i ] q α n q i = 1 k ( h - i ) x i d μ - q ( x 1 ) d μ - q ( x k ) = - p p w x 1 + + x k [ x + i = 1 k x i ] q α n q i = 1 k ( h - i ) x i d μ - q ( x 1 ) d μ - q ( x k ) + [ 2 ] q p p w x 2 + + x k [ x + i = 2 k x i ] q α n q i = 2 k ( h - i ) x i d μ - q ( x 2 ) d μ - q ( x k ) .
(20)

Therefore, by (3) and (20), we obtain the following theorem.

Theorem 9. For h , α and n +, we have
w q h n , q , w ( α ) ( h , k | x + 1 ) + n , q , w ( α ) ( h , k | x ) = [ 2 ] q n , q , w ( α ) ( h - 1 , k - 1 | x ) .
Note that
q α x p p w x 1 + + w k [ x + i = 1 k x i ] q α n q i = 1 k ( h i ) x i d μ q ( x 1 ) d μ q ( x k ) = ( q α 1 ) p p w x 1 + + x k [ x + i = 1 k x i ] q α n + 1 q i = 1 k ( h α i ) x i d μ q ( x 1 ) d μ q ( x k ) + p p w x 1 + + x k [ x + i = 1 k x i ] q α n q i = 1 k ( h α i ) x i d μ q ( x 1 ) d μ q ( x k ) = ( q α 1 ) E ˜ n + 1 , q , w ( α ) ( h α , k | x ) + E ˜ n , q , w ( α ) ( h α , k | x ) .
(21)

Therefore, by (21), we obtain the following theorem.

Theorem 10. For n +, we have
q α x n , q , w ( α ) ( h , k | x ) = ( q α - 1 ) n + 1 , q , w ( α ) ( h - α , k | x ) + n , q , w ( α ) ( h - α , k | x ) .
Let d with d ≡ 1 (mod 2). Then we get
p p w x 1 + + x k [ x + j = 1 k x j ] q α n q j = 1 k ( h j ) x j d μ q ( x 1 ) d μ q ( x k ) = [ d ] q α n [ d ] q k a 1 , , a k = 0 d 1 w a 1 + + a k q h j = 1 k a j j = 2 k ( j 1 ) a j ( 1 ) j = 1 k a j × p p w d ( x 1 + + x k ) [ x + j = 1 k a j d + j = 1 k x j ] q α d n q d j = 1 k ( h j ) x j d μ q d ( x 1 ) d μ q d ( x k ) .
(22)

Therefore, by (22), we obtain the following theorem.

Theorem 11. For d with d ≡ 1 (mod 2), we have
n , q , w ( α ) ( h , k | d x ) = [ d ] q α n [ d ] - q k a 1 , , a k = 0 d - 1 w a 1 + + a k q h j = 1 k a j - j = 2 k ( j - 1 ) a j ( - 1 ) j = 1 k a j n , q d , w d ( α ) h , k | x + a 1 + + a k d .
Let n , q , w ( α ) ( k , k | x ) = n , q , w ( α ) ( k | x ) . Then we get
( q α - 1 ) n n , q , w ( α ) ( k | x ) = l = 0 n n l ( - 1 ) n - l q α l x [ 2 ] q k ( 1 + w q α l + k ) ( 1 + w q α l + 1 ) ,
and
p p w - ( x 1 + + x k ) [ k - x + i = 1 k x i ] q - α n q - i = 1 k ( k - i ) x i d μ - q - 1 ( x 1 ) d μ - q - 1 ( x k ) = q k 2 ( 1 - q - α ) n [ 2 ] q k l = 0 n n l ( - 1 ) l q α l x 1 ( 1 + w q α l + 1 ) ( 1 + w q α l + k ) = ( - 1 ) n q n α q k 2 [ 2 ] q k ( 1 - q α ) n l = 0 n n l ( - 1 ) l q α l x ( 1 + w q α l + 1 ) ( 1 + w q α l + k ) = ( - 1 ) n q α n + k 2 n , q , w ( α ) ( k | x ) .
(23)

Therefore, by (23), we obtain the following theorem.

Theorem 12. For n +, we have
n , q - 1 , w - 1 ( α ) ( k | k - x ) = ( - 1 ) n w k q α n + k 2 n , q , w ( α ) ( k | x ) .
Let x = k in Theorem 12. Then we see that
n , q - 1 , w - 1 ( α ) ( k | 0 ) = ( - 1 ) n w k q α n + k 2 n , q , w ( α ) ( k | k ) .
(24)
From (15), we note that
w q k n , q , w ( α ) ( k | x + 1 ) + n , q , w ( α ) ( k | x ) = [ 2 ] q n , q , w ( α ) ( k - 1 | x ) .
(25)
It is easy to show that
( q α - 1 ) n n , q , w ( α ) ( k | 0 ) = l = 0 n n l ( - 1 ) l + n [ 2 ] q k ( 1 + w q α l + 1 ) ( 1 + w q α l + k ) .
By simple calculation, we get
l = 0 n n l ( q α - 1 ) l p p w i = 1 k x k [ i = 1 k x k ] q α l q l = i k ( k - i ) x i d μ - q ( x 1 ) d μ - q ( x k ) = [ 2 ] q k ( 1 + w q α n + k ) ( 1 + w q α n + k - 1 ) ( 1 + w q α n + 1 ) .
(26)
From (26), we note that
l = 0 n n l ( q α - 1 ) l l , q , w ( α ) ( k | 0 ) = [ 2 ] q k ( 1 + w q α n + k ) ( 1 + w q α n + k - 1 ) ( 1 + w q α n + 1 ) ,
and
n , q , w ( α ) ( k | x ) = p p w i = 1 k x k [ x + i = 1 k x k ] q α n q i = 1 k ( k - i ) x i d μ - q ( x 1 ) d μ - q ( x k ) = l = 0 n n l [ x ] q α n - l q α l x l , q , w ( α ) ( k | 0 ) = q x α q , w ( α ) ( k | 0 ) + [ x ] q α n , n + ,

with the usual convention about replacing ( q , w ( α ) ( k | 0 ) ) n by n , q , w ( α ) ( k | 0 ) .

Put x = 0 in (25), we get
w q k n , q , w ( α ) ( k | 1 ) + n , q , w ( α ) ( k | 0 ) = [ 2 ] q n , q ( α ) ( k - 1 | 0 ) .
Thus, we have
w q k ( q α q , w ( α ) ( k | 0 ) + 1 ) n + n , q , w ( α ) ( k | 0 ) = [ 2 ] q n , q , w ( α ) ( k - 1 | 0 ) ,

with the usual convention about replacing ( q , w ( α ) ( k | 0 ) ) n by n , q , w ( α ) ( k | 0 ) .

Declarations

Acknowledgements

The authors express their gratitude to the referee for his/her valuable comments.

Authors’ Affiliations

(1)
Department of Mathematics, Hannam University

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© Lee et al; licensee Springer. 2012

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