Some new identities on the twisted carlitz's q-bernoulli numbers and q-bernstein polynomials

  • Lee-Chae Jang
  • Taekyun Kim
  • Young-Hee Kim
  • Byungje Lee
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Abstract

In this paper, we consider the twisted Carlitz's q-Bernoulli numbers using p-adic q-integral on ℤ p . From the construction of the twisted Carlitz's q-Bernoulli numbers, we investigate some properties for the twisted Carlitz's q-Bernoulli numbers. Finally, we give some relations between the twisted Carlitz's q-Bernoulli numbers and q-Bernstein polynomials.

Keywords

q-Bernoulli numbers p-adic q-integral twisted 

1. Introduction and preliminaries

Let p be a fixed prime number. Throughout this paper, ℤ p , p Open image in new window and p Open image in new window will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of p Open image in new window, respectively. Let ℕ be the set of natural numbers, and let ℤ+ = ℕ ∪ {0}. Let ν p be the normalized exponential valuation of p Open image in new window with | p | p = p - ν p ( p ) = 1 p Open image in new window. In this paper, we assume that q p Open image in new window with |1 - q| p < 1. The q-number is defined by [ x ] q = 1 - q x 1 - q Open image in new window. Note that limq → 1[x] q = x.

We say that f is a uniformly differentiable function at a point a ∈ ℤ p , and denote this property by fUD(ℤ p ), if the difference quotient F f ( x , y ) = f ( x ) - f ( y ) x - y Open image in new window has a limit f'(a) as (x, y) → (a, a). For fUD(ℤ p ), the p-adic q-integral on ℤ p , which is called the q-Volkenborn integral, is defined by Kim as follows:
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 , ( see  [ 1 ] ) . Open image in new window
(1)
In [2], Carlitz defined q-Bernoulli numbers, which are called the Carlitz's q-Bernoulli numbers, by
β 0 , q = 1 , and q ( q β + 1 ) n - β n , q = 1 if n = 1 , 0 if n > 1 , Open image in new window
(2)

with the usual convention about replacing β n by βn, q.

In [2, 3], Carlitz also considered the expansion of q-Bernoulli numbers as follows:
β 0 , q ( h ) = h [ h ] q , and q h ( q β ( h ) + 1 ) n - β n , q ( h ) = 1 if n = 1 , 0 if n > 1 , Open image in new window
(3)

with the usual convention about replacing (β(h)) n by β n , q ( h ) Open image in new window.

Let C p n = { ξ | ξ p n = 1 } Open image in new window be the cyclic group of order p n , and let T p = lim n C p n = C p = n 0 C p n Open image in new window (see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]). Note that T p is a locally constant space.

For ξ ∈ T p , the twisted q-Bernoulli numbers are defined by
t ξ e t - 1 = e B ξ t = n = 0 B n , ξ t n n ! , Open image in new window
(4)
(see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]). From (4), we note that
B 0 , q = 0 , and ξ ( B ξ + 1 ) n - B n , ξ = 1 if n = 1 , 0 if n > 1 , Open image in new window
(5)

with the usual convention about replacing B ξ n Open image in new window by B n,ξ (see [17, 18, 19]). Recently, several authors have studied the twisted Bernoulli numbers and q-Bernoulli numbers in the area of number theory(see [17, 18, 19]).

In the viewpoint of (5), it seems to be interesting to investigate the twisted properties of (3). Using p-adic q-integral equation on ℤ p , we investigate the properties of the twisted q-Bernoulli numbers and polynomials related to q-Bernstein polynomials. From these properties, we derive some new identities for the twisted q-Bernoulli numbers and polynomials. Final purpose of this paper is to give some relations between the twisted Carlitz's q-Bernoulli numbers and q-Bernstein polynomials.

2. On the twisted Carlitz 's q-Bernoulli numbers

In this section, we assume that n ∈ ℤ+, ξ ∈ T p and q p Open image in new window with |1 - q| p < 1.

Let us consider the n th twisted Carlitz's q-Bernoulli polynomials using p-adic q-integral on ℤ p as follows:
β n , ξ , q ( x ) = p [ y + x ] q n ξ y d μ q ( y ) (1) = 1 ( 1 - q ) n l = 0 n n l ( - 1 ) l q l x p ξ y q l y d μ q ( y ) (2) = 1 ( 1 - q ) n - 1 l = 0 n n l l + 1 1 - ξ q l + 1 ( - 1 ) l q l x . (3) (4) Open image in new window
(6)

In the special case, x = 0, β n,ξ,q (0) = β n,ξ,q are called the n th twisted Carlitz's q-Bernoulli numbers.

From (6), we note that
β n , ξ , q ( x ) = 1 ( 1 - q ) n - 1 l = 0 n - 1 n l ( - 1 ) l q l x 1 1 - ξ q l + 1 (1) + 1 ( 1 - q ) n - 1 l = 0 n n l ( - 1 ) l q l x 1 1 - ξ q l + 1 (2) = - n m = 0 ξ m q 2 m + x [ x + m ] q n - 1 + m = 0 ξ m q m ( 1 - q ) [ x + m ] q n . (3) (4) Open image in new window
(7)

Therefore, by (7), we obtain the following theorem.

Theorem 1. For n ∈ ℤ+, we have
β n , ξ , q ( x ) = - n m = 0 ξ m q m [ x + m ] q n - 1 + ( 1 - q ) ( n + 1 ) m = 0 ξ m q m [ x + m ] q n . Open image in new window
Let F q, ξ (t, x) be the generating function of the twisted Carlitz's q-Bernoulli poly-nomials, which are given by
F q , ξ ( t , x ) = e β ξ , q ( x ) t = n = 0 β n , ξ , q ( x ) t n n ! , Open image in new window
(8)

with the usual convention about replacing (β ξ,q (x)) n by β n,ξ,q (x).

By (8) and Theorem 1, we get
F q , ξ ( t , x ) = n = 0 β n , ξ , q ( x ) t n n ! (1) = - t m = 0 ξ m q 2 m + x e [ x + m ] q t + ( 1 - q ) m = 0 ξ m q m e [ x + m ] q t . (2) (3) Open image in new window
(9)
Let Fq,ξ(t, 0) = F q,ξ (t). Then, we have
q ξ F q , ξ ( t , 1 ) - F q , ξ ( t ) = t + ( q - 1 ) . Open image in new window
(10)

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

Theorem 2. For n ∈ ℤ+, we have
β 0 , ξ , q ( x ) = q - 1 q ξ - 1 , a n d q ξ β n , ξ , q ( 1 ) - β n , ξ , q = 1 i f n = 1 , 0 i f n > 1 . Open image in new window
From (6), we note that
β n , ξ , q ( x ) = l = 0 n n l [ x ] q n - l q l x p ξ y [ y ] q l d μ q ( y ) (1) = l = 0 n n l [ x ] q n - l q l x β l , ξ , q (2) = [ x ] q + q x β ξ , q n , (3) (4) Open image in new window
(11)
with the usual convention about replacing (βξ,q) n by βn,ξ,q. By (11) and Theorem 2, we get
q ξ ( q β ξ , q + 1 ) n - β n , ξ , q = q - 1 if n = 0 , 1 if n = 1 , 0 if n > 1 . Open image in new window
(12)
It is easy to show that
β n , ξ - 1 , q - 1 ( 1 - x ) = p ξ - y [ 1 - x + y ] q - 1 n d μ q - 1 ( y ) (1)  = ( - 1 ) n q n ( 1 - q ) n l = 0 n n l ( - 1 ) l q - l + l x p ξ - y q - l y d μ q - 1 ( y ) (2)  = ξ q n ( - 1 ) n 1 ( 1 - q ) n - 1 l = 0 n n l ( - 1 ) l q l x ( l + 1 1 - ξ q l + 1 ) (3)  = ξ q n ( - 1 ) n β n , ξ , q ( x ) . (4)  (5)  Open image in new window
(13)

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

Theorem 3. For n ∈ ℤ+, we have
β n , ξ - 1 , q - 1 ( 1 - x ) = ξ q n ( - 1 ) n β n , ξ , q ( x ) . Open image in new window
From Theorem 3, we can derive the following functional equation:
F q - 1 , ξ - 1 ( t , 1 - x ) = ξ F q , ξ ( - q t , x ) . Open image in new window
(14)

Therefore, by (14), we obtain the following corollary.

Corollary 4. Let F q , ξ ( t , x ) = n = 0 β n , ξ , q ( x ) t n n ! Open image in new window. Then we have
F q - 1 , ξ - 1 ( t , 1 - x ) = ξ F q , ξ ( - q t , x ) . Open image in new window
By (11), we get that
q 2 ξ 2 β n , ξ , q ( 2 ) = q 2 ξ 2 l = 0 n n l q l ( 1 + q β ξ , q ) l (1) = q 2 ξ 2 ( 1 - q 1 - q ξ ) + n 1 q 2 ξ ( 1 + β 1 , ξ , q ) + q 2 ξ 2 l = 0 n n l q l β l , ξ , q ( 1 ) (2) = ( 1 - q ) q 2 ξ 2 1 - q ξ + n 1 q 2 ξ + q ξ l = 0 n n l q l β l , ξ , q (3) = 1 - q 1 - q ξ q 2 ξ 2 + n q 2 ξ - q ξ 1 - q 1 - q ξ + β n , ξ , q , if n > 1 . (4) (5)  Open image in new window
(15)

Therefore, by (15), we obtain the following theorem.

Theorem 5. For n ∈ ℕ with n > 1, we have
β n , ξ , q ( 2 ) = 1 - q 1 - q ξ + n ξ - 1 q ξ ( 1 - q 1 - q ξ ) + ( 1 q ξ ) 2 β n , ξ , q . Open image in new window
By a simple calculation, we easily set
ξ p [ 1 - x ] q - 1 n ξ x d μ q ( x ) = ξ ( - 1 ) n q n p [ x - 1 ] q n ξ x d μ q ( x ) (1) = ξ ( - 1 ) n q n β n , ξ , q ( - 1 ) = β n , ξ - 1 , q - 1 ( 2 ) . (2) (3) Open image in new window
(16)
For n ∈ ℤ+ with n > 1, we have
ξ p [ 1 - x ] q - 1 n ξ x d μ q ( x ) = β n , ξ - 1 , q - 1 ( 2 ) (1) = ξ ( 1 - q 1 - q ξ ) + n ξ - q ξ 2 ( 1 - q 1 - q ξ ) + ( q ξ ) 2 β n , ξ - 1 , q - 1 (2) = ξ ( 1 - q ) + n ξ + ( q ξ ) 2 β n , ξ - 1 , q - 1 . (3) (4) Open image in new window
(17)

Therefore, by (16) and (17), we obtain the following theorem.

Theorem 6. For n ∈ ℤ+with n > 1, we have
p [ 1 - x ] q - 1 n ξ x d μ q ( x ) = ( 1 - q ) + n + q 2 ξ β n , ξ - 1 , q - 1 . Open image in new window
For x ∈ℤ p and n, k ∈ ℤ+, the p-adic q-Bernstein polynomials are given by
B k , n ( x , q ) = ( n k ) [ x ] q k [ 1 x ] q 1 n k , Open image in new window
(18)

(see [8, 20]).

In [8], the q-Bernstein operator of order n is given by
B n , q ( f | x ) = k = 0 n f ( n k ) B k , n ( x , q ) = k = 0 n f ( n k ) n k [ x ] q k [ 1 - x ] q - 1 n - k . Open image in new window

Let f be continuous function on ℤ p . Then, the sequence B n , q ( f | x ) Open image in new window converges uniformly to f on ℤ p (see [8]). If q is same version in (18), we cannot say that the sequence B n , q ( f | x ) Open image in new window converges uniformly to f on ℤ p .

Let s ∈ ℕ with s ≥ 2. For n1, ..., n s , k ∈ ℤ+ with n1 + · · · + n s > sk + 1, we take the p-adic q-integral on ℤ p for the multiple product of q-Bernstein polynomials as follows:
p ξ x B k , n 1 ( x , q ) B k , n s ( x , q ) d μ q ( x ) = ( n 1 k ) ( n s k ) p [ x ] q k [ 1 x ] q 1 n 1 + + n s s k ξ x d μ q ( x ) = ( n 1 k ) ( n s k ) l = 0 s k ( s k l ) ( 1 ) l + s k p [ 1 x ] q 1 n 1 + + n s l ξ x d μ q ( x ) = ( n 1 k ) ( n s k ) l = 0 s k ( s k l ) ( 1 ) l + s k × ( q 2 ξ β n 1 + + n s l , ξ 1 , q 1 + n 1 + + n s l + 1 q ) d μ q ( x ) = { q 2 ξ β n 1 + + n s , ξ 1 , q 1 + n 1 + + n s + ( 1 q ) if k = 0 , q 2 ξ ( k n 1 ) ( k n s ) l = 0 s k ( l s k ) ( 1 ) l + s k β n 1 + + n s l , ξ 1 . q 1 if k > 0 , Open image in new window
(19)
and we also have
p ξ x B k , n 1 ( x , q ) B k , n s ( x , q ) d μ q ( x ) = n 1 k n s k l = 0 n 1 + + n s - s k n 1 + + n s - s k l ( - 1 ) l β l + s k , ξ , q . Open image in new window
(20)

By comparing the coefficients on the both sides of (19) and (20), we obtain the following theorem.

Theorem 7. Let s ∈ ℕ with s ≥ 2. For n1, ..., n s , k ∈ ℤ+with n1 + ⋯ + n s > sk + 1, we have
l = 0 n 1 + + n s s k ( n 1 + + n s s k l ) ( 1 ) l β l + s k , ξ , q = { q 2 ξ β n 1 + + n s , ξ 1 , q 1 + n 1 + + n s + ( 1 q ) i f k = 0 , q 2 ξ l = 0 s k ( l s k ) ( 1 ) l + s k β n 1 + + n s l , ξ 1 . q 1 i f k > 0 . Open image in new window

Notes

Acknowledgements

The authors express their sincere gratitude to referees for their valuable suggestions and comments. This paper was supported by the research grant Kwangwoon University in 2011.

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© Kim et al; licensee Springer. 2011

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.

Authors and Affiliations

  • Lee-Chae Jang
    • 1
  • Taekyun Kim
    • 2
  • Young-Hee Kim
    • 2
  • Byungje Lee
    • 3
  1. 1.Department of Mathematics and Computer ScienceKonkuk UniversityChungjuRepublic of Korea
  2. 2.Division of General Education-MathematicsKwangwoon UniversitySeoulRepublic of Korea
  3. 3.Department of Wireless Communications EngineeringKwangwoon UniversitySeoulRepublic of Korea

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