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

Journal of Inequalities and Applications20112011:52

https://doi.org/10.1186/1029-242X-2011-52

Received: 21 February 2011

Accepted: 13 September 2011

Published: 13 September 2011

Abstract

In this paper, we consider the twisted Carlitz's q-Bernoulli numbers using p-adic q-integral on 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 numbersp-adic q-integraltwisted

1. Introduction and preliminaries

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

We say that f is a uniformly differentiable function at a point a p , and denote this property by f UD( p ), if the difference quotient F f ( x , y ) = f ( x ) - f ( y ) x - y has a limit f'(a) as (x, y) → (a, a). For f UD( p ), the p-adic q-integral on 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 ] ) .
(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 ,
(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 ,
(3)

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

Let C p n = { ξ | ξ p n = 1 } be the cyclic group of order p n , and let T p = lim n C p n = C p = n 0 C p n (see [116]). 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 ! ,
(4)
(see [119]). 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 ,
(5)

with the usual convention about replacing B ξ n by B n,ξ (see [1719]). Recently, several authors have studied the twisted Bernoulli numbers and q-Bernoulli numbers in the area of number theory(see [1719]).

In the viewpoint of (5), it seems to be interesting to investigate the twisted properties of (3). Using p-adic q-integral equation on 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 +, ξ T p and q p with |1 - q| p < 1.

Let us consider the n th twisted Carlitz's q-Bernoulli polynomials using p-adic q-integral on 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)
(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)
(7)

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

Theorem 1. For n +, 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 .
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 ! ,
(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)
(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 ) .
(10)

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

Theorem 2. For n +, 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 .
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)
(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 .
(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) 
(13)

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

Theorem 3. For n +, we have
β n , ξ - 1 , q - 1 ( 1 - x ) = ξ q n ( - 1 ) n β n , ξ , q ( x ) .
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 ) .
(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 ! . Then we have
F q - 1 , ξ - 1 ( t , 1 - x ) = ξ F q , ξ ( - q t , x ) .
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) 
(15)

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

Theorem 5. For 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 .
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)
(16)
For n + 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)
(17)

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

Theorem 6. For n +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 .
For x p and n, k +, 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 ,
(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 .

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

Let s with s ≥ 2. For n1, ..., n s , k + with n1 + · · · + n s > sk + 1, we take the p-adic q-integral on 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 ,
(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 .
(20)

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

Theorem 7. Let s with s ≥ 2. For n1, ..., n s , k +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 .

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Konkuk University
(2)
Division of General Education-Mathematics, Kwangwoon University
(3)
Department of Wireless Communications Engineering, Kwangwoon University

References

  1. 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.1999.2373MATHMathSciNetView ArticleGoogle Scholar
  2. Carlitz L: q -Bernoulli numbers and polynomials. Duke Math J 1948, 15: 987–1000. 10.1215/S0012-7094-48-01588-9MATHMathSciNetView ArticleGoogle Scholar
  3. Kim T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ J Math Phys 2008, 15: 51–57.MATHMathSciNetView ArticleGoogle Scholar
  4. Bernstein S: Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilities. Commun Soc Math Kharkow 1912, 13: 1–2.Google Scholar
  5. Cangul IN, Kurt V, Ozden H, Simsek Y: On the higher-order w - q -Genocchi numbers. Adv Stud Contemp Math 2009, 19: 39–57.MathSciNetGoogle Scholar
  6. Govil NK, Gupta V: Convergence of q -Meyer-König-Zeller-Durrmeyer operators. Adv Stud Contemp Math 2009, 19: 97–108.MATHMathSciNetGoogle Scholar
  7. Jang L-C: A study on the distribution of twisted q -Genocchi polynomials. Adv Stud Contemp Math 2009, 19: 181–189.Google Scholar
  8. Kim T: A note on q -Bernstein polynomials. Russ J Math Phys 2011, 18: 41–50.View ArticleGoogle Scholar
  9. Kim T: q -Volkenborn integration. Russ J Math Phys 2002, 9: 288–299.MATHMathSciNetGoogle Scholar
  10. Kim T, Choi J, Kim Y-H: Some identities on the q -Bernstein polynomials, q -Stirling numbers and q -Bernoulli numbers. Adv Stud Contemp Math 2010, 20: 335–341.MATHGoogle Scholar
  11. Kim T: Barnes type multiple q -zeta functions and q -Euler polynomials. J Physics A: Math Theor 2010, 43: 11. 255201Google Scholar
  12. Kurt V: further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials. Appl Math Sci (Ruse) 2008, 3: 2757–2764.MathSciNetGoogle Scholar
  13. Rim S-H, Moon E-J, Lee S-J, Jin J-H: Multivariate twisted p -adic q -integral on p associated with twisted q -Bernoulli polynomials and numbers. J Inequal Appl 2010, 2010: Art ID 579509. 6 ppMathSciNetView ArticleGoogle Scholar
  14. Ryoo CS, Kim YH: A numericla investigation on the structure of the roots of the twisted q -Euler polynomials. Adv Stud Contemp Math 2009, 19: 131–141.MATHMathSciNetGoogle Scholar
  15. Ryoo CS: On the generalized Barnes' type multiple q -Euler polynomials twisted by ramified roots of unity. Proc Jangjeon Math Soc 2010, 13: 255–263.MATHMathSciNetGoogle Scholar
  16. Ryoo CS: A note on the weighted q-Euler numbers and polynomials. Adv Stud Contemp Math 2011, 21: 47–54.MATHMathSciNetGoogle Scholar
  17. Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv Stud Contemp Math 2008, 16: 251–278.MATHMathSciNetGoogle Scholar
  18. Kim T: Non-Archimedean q -integrals associated with multiple Changhee q -Bernoulli polynomials. Russ J Math Phys 2003, 10: 91–98.MATHMathSciNetGoogle Scholar
  19. Simsek Y: Theorems on twisted L -function and twisted Bernoulli numbers. Adv Stud Contemp Math 2005, 11: 205–218.MATHMathSciNetGoogle Scholar
  20. Bayad A, Kim T: Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials. Russ J Math Phys 2011, 18: 133–143. 10.1134/S1061920811020014MATHMathSciNetView ArticleGoogle Scholar

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© Kim et al; licensee Springer. 2011

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