Skip to content

Advertisement

  • Research
  • Open Access

Some identities on the twisted q-Bernoulli numbers and polynomials with weight α

Journal of Inequalities and Applications20122012:176

https://doi.org/10.1186/1029-242X-2012-176

  • Received: 24 April 2012
  • Accepted: 1 August 2012
  • Published:

Abstract

In this paper, we consider the twisted q-Bernoulli numbers and polynomials with weight α by using the bosonic q-integral on . From the construction of the twisted q-Bernoulli numbers with weight α, we derive some identities and relations.

MSC: 11B68, 11S40, 11S80.

Keywords

  • Bernoulli numbers and polynomials
  • bosonic q-integral
  • twisted q-Bernoulli numbers and polynomials with weight α

1 Introduction

Let p be a fixed prime number. Throughout this paper, , , and will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the p-adic completion of the algebraic closure of , respectively. Let ν p be the normalized exponential valuation of with | p | p = p ν p ( p ) = p 1 . When one talks of q-extension, q is variously considered as an indeterminate, a complex number , or a p-adic number . If , one normally assume | q | < 1 . If , then we assume that | 1 q | p < p 1 1 p , so that q x = exp ( x log q ) for | x | p 1 . The q-number is defined by [ x ] q = 1 q x 1 q (see [125]). Note that lim q 1 [ x ] q = x .

We say that f is uniformly differentiable function at a point , and denote this property by , 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 , the p-adic q-integral on , which is called the bosonic q-integral, defined by Kim as follows:
(1)
From (1), we note that
(2)

where f 1 = sup { | f ( 0 ) | p , sup x y | f ( x ) f ( y ) x y | p } .

In [3], Carlitz defined q-Bernoulli numbers which are called the Carlitz’s q-Bernoulli numbers, by
β 0 , q = 1 and q ( q β q + 1 ) n β n , q = { 1 if  m = 1 , 0 if  m > 1 ,
(3)
with the usual convention about replacing ( β q ( h ) ) n by β n , q ( h ) . In [3], Carlitz also considered the expansion of q-Bernoulli numbers as follows:
β 0 , q ( h ) = h [ h ] q and q h ( q β q ( h ) + 1 ) n β n , q ( h ) = { 1 if  n = 1 , 0 if  n > 1 ,
(4)
with the usual convention about replacing ( β q ( h ) ) n by β n , q ( h ) . In [15], for and , q-Bernoulli numbers with weight α is defined by Kim as follows:
β 0 , q ( α ) and q ( q α β ˜ q ( α ) + 1 ) n β ˜ n , q ( α ) = { α [ α ] q if  n = 1 , 0 if  n > 1 ,
(5)

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

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 = n 0 C p n (see [8, 13, 2024]). Note that T p is a locally constant space. For ξ T p , the twisted Bernoulli numbers are defined by
t ξ e t 1 = e B ξ t = n = 0 B n , ξ t n n ! ,
(6)

with the usual convention about replacing ( B ξ ) n by B n , q (see [15, 1924]).

In the view point of (6), we will try to study the twisted q-Bernoulli numbers with weight α. By using the p-adic q-integral on , we give some identities and relations on the twisted q-Bernoulli numbers and polynomials with weight α.

2 The twisted q-Bernoulli numbers and polynomials with weight α

Let f n ( x ) = f ( x + n ) . In [[15], Theorem 1], Kim proved the following integral equation:
q n I q ( f n ) I q ( f ) = ( q 1 ) l = 0 n 1 q l f ( l ) + q 1 log q l = 0 n 1 q l f ( l ) .
(7)
In particular, when n = 1 , we have
q I q ( f 1 ) I q ( f ) = ( q 1 ) f ( 0 ) + q 1 log q f ( 0 ) .
(8)
Let and . The n th q-Bernoulli polynomials with weight α are defined by
(9)
Then, by using the bosonic p-adic q-integral on , we evaluate the above equation (9) as follows:
(10)

In the special case x = 0 , β ˜ n , ξ , q α ( 0 ) = β ˜ n , ξ , q α are called the n th twisted q-Bernoulli numbers with weight α.

From (10), we note that
(11)
From (11), when x = 0 , we get
β ˜ n , ξ , q α = n α [ α ] q m = 0 ξ m q m α + m [ m ] q α n 1 + ( 1 q ) m = 0 ξ m q m [ m ] q α n .
(12)

Therefore, by (10) and (12), we obtain the following theorem.

Theorem 1 Let and . Then we

When x = 0 , we can obtain some identity on the twisted q-Bernoulli numbers with weight α as follows.

Corollary 2 For and . Then we have
β ˜ n , ξ , q ( α ) n = α [ α ] q m = 0 ξ m q m α + m [ m ] q α n 1 + 1 q n m = 0 ξ m q m [ m ] q α n .
For α = 1 , we note that β ˜ n , ξ , q ( 1 ) ( x ) are the twisted Carlitz’s q-Bernoulli polynomials and β ˜ n , ξ , q ( 1 ) are the twisted Carlitz’s q-Bernoulli numbers. By Corollary 2, we easily get
(13)
From (13), we have
n = 0 β ˜ n , ξ , q ( α ) t n n ! = t α [ α ] q m = 0 ξ m q ( m + 1 ) α + m e [ m ] q α t + ( 1 q ) m = 0 ξ m q m e [ m ] q α t .
(14)

Therefore, we obtain the following corollary.

Corollary 3 Let and F ξ , q ( α ) ( t ) = n = 0 β ˜ n , ξ , q ( α ) t n n ! . Then we have
F ξ , q ( α ) ( t ) = t α [ α ] q m = 0 ξ m q ( m + 1 ) α + m e [ m ] q α t + ( 1 q ) m = 0 ξ m q m e [ m ] q α t .
Let F ξ , q ( α ) ( t , x ) = n = 0 n = 0 β ˜ n , ξ , q ( α ) ( x ) t n n ! . From Theorem 1, we have
(15)
By simple calculation, we easily get
(16)

Therefore, by (12), (15), and (16), we obtain the following theorem.

Theorem 4 For and , we have
β ˜ n , ξ , q ( α ) ( x ) = 1 q ( 1 q α ) n l = 0 n ( n l ) ( 1 ) l q α l x ( α l + 1 1 ξ q α l + 1 ) = n α [ α ] q m = 0 ξ m q ( m + 1 ) α + m [ m + x ] q α n 1 + ( 1 q ) m = 0 ξ m q m [ m + x ] q α n .
(17)

Moreover, β ˜ n , ξ , q ( α ) ( x ) = l = 0 n ( n l ) [ x ] q α n l q α l x β ˜ l , ξ , q ( α ) .

By (7), we see that
(18)

Therefore, we obtain the following theorem.

Theorem 5 For , , and , we have
If we take f ( x ) = ξ x e [ x ] q α t , by (8), then we have
(19)

Therefore, by Theorem 5, we obtain the following theorem.

Theorem 6 For and , we have
ξ q β ˜ 0 , ξ , q ( α ) ( 1 ) β ˜ 0 , ξ , q ( α ) = { q 1 if n = 0 , α [ α ] q if n = 1 , 0 if n > 1 .
Remark that when n = 0 , we have β ˜ 0 , ξ , q ( α ) = q 1 ξ q 1 . By (16) and Theorem 6, we get
β ˜ n , ξ , q ( α ) ( x ) = l = 0 n ( n l ) [ x ] q α n l q α l x β ˜ l , ξ , q ( α ) = ( [ x ] q α + q α x β ˜ ξ , q ( α ) ) n ,
(20)

with the usual convention about replacing ( β ˜ ξ , q ( α ) ) n by β ˜ n , ξ , q ( α ) . By (20) and Theorem 6, we obtain the following theorem.

Theorem 7 For and , we have
ξ q ( q α β ˜ ξ , q ( α ) + 1 ) n β ˜ n , ξ , q ( α ) = { q 1 if n = 0 , α [ α ] q if n = 1 , 0 if n > 1 ,

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

From (8), we can easily derive the following equation (21). For , we get
(21)

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

Theorem 8 For , , and , we have
β ˜ n , ξ , q ( α ) ( x ) = [ d ] q α n [ d ] q a = 0 d 1 q a β ˜ n , ξ , q d ( α ) ( x + a d ) .
From (9), we note that
(22)

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

Theorem 9 For and , we have
β ˜ n , ξ 1 , q 1 ( α ) ( 1 x ) = ( 1 ) n q α n β ˜ n , ξ , q ( α ) ( x ) .
It is easy to show that
(23)

By (22) and (23), we obtain the following corollary.

Corollary 10 For and , we have

Declarations

Acknowledgement

This paper was supported by Kon-Kuk University in 2012.

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Kon-Kuk University, Chungju, 138-701, Korea

References

  1. Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20: 389–401.MathSciNetMATHGoogle Scholar
  2. Bayad A, Kim T: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 2010, 20: 247–253.MathSciNetMATHGoogle Scholar
  3. Carlitz L: Expansion of q -Bernoulli numbers and polynomials. Duke Math. J. 1958, 25: 355–364. 10.1215/S0012-7094-58-02532-8MathSciNetView ArticleMATHGoogle Scholar
  4. Carlitz L: q -Bernoulli numbers and polynomials. Duke Math. J. 1948, 15: 987–1000. 10.1215/S0012-7094-48-01588-9MathSciNetView ArticleMATHGoogle Scholar
  5. Cancul IN, Kurt V, Ozden H, Simsek Y: On the higher-order w - q -Genocchi numbers. Adv. Stud. Contemp. Math. 2009, 9: 39–57.MathSciNetMATHGoogle Scholar
  6. Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20: 7–21.MathSciNetMATHGoogle Scholar
  7. Hegazi AS, Mansour M: A note on q -Bernoulli numbers and polynomials. J. Nonlinear Math. Phys. 2006, 13: 9–18.MathSciNetView ArticleMATHGoogle Scholar
  8. Jang LC, Kim WJ, Simsek Y: A study on the p -adic integral representation on associated with Bernstein and Bernoulli polynomials. Adv. Differ. Equ. 2002., 2002: Article ID 163217Google Scholar
  9. Jang LC: A study on the distribution of twisted q -Genocchi polynomials. Adv. Stud. Contemp. Math. 2009, 19: 181–189.MathSciNetMATHGoogle Scholar
  10. Kim T, Choi J: On the q -Bernoulli numbers and polynomials with weight α . Abstr. Appl. Anal. 2011., 2011: Article ID 392025Google Scholar
  11. Kim T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 2008, 15: 51–57.MathSciNetView ArticleMATHGoogle Scholar
  12. Kim T: Some identities on the q -Euler polynomials of higher order and q -Stirling numbers by the fermionic p -adic integral on . Russ. J. Math. Phys. 2009, 16: 484–491. 10.1134/S1061920809040037MathSciNetView ArticleMATHGoogle Scholar
  13. Kim T: Barnes-type multiple q -zeta functions and q -Euler polynomials. J. Phys. A, Math. Theor. 2010., 43: Article ID 255201Google Scholar
  14. 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 ArticleMATHGoogle Scholar
  15. Kim T: On the weighted q -Bernoulli numbers and polynomials. Adv. Stud. Contemp. Math. 1999, 21(2):207–215.MathSciNetMATHGoogle Scholar
  16. Kupershmidt BA: Reflection symmetries of q -Bernoulli polynomials. J. Nonlinear Math. Phys. 2005, 12: 412–422. 10.2991/jnmp.2005.12.s1.34MathSciNetView ArticleGoogle Scholar
  17. Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18: 41–48.MathSciNetMATHGoogle Scholar
  18. Rim S-H, Jin J-H, Moon E-J, Lee S-J: On multiple interpolation function of the q -Genocchi polynomials. J. Inequal. Appl. 2010., 2010: Article ID 351419Google Scholar
  19. Rim S-H, Moon E-J, Lee S-J, Jin J-H: Multivariate twisted p -adic q -integral on associated with twisted q -Bernoulli polynomials and numbers. J. Inequal. Appl. 2010., 2010: Article ID 579509Google Scholar
  20. Ryoo CS: On the generalized Barnes type multiple q -Euler polynomials twisted by ramified roots of unity. Proc. Jangjeon Math. Soc. 2010, 13(2):255–263.MathSciNetMATHGoogle Scholar
  21. Ryoo CS: A note on the weighted q -Euler numbers and polynomials. Adv. Stud. Contemp. Math. 2011, 21: 47–54.MathSciNetMATHGoogle Scholar
  22. Simsek Y: Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions. Adv. Stud. Contemp. Math. 2008, 16: 251–278.MathSciNetMATHGoogle Scholar
  23. Simsek Y: On p -adic twisted q - L -functions related to generalized twisted Bernoulli numbers. Russ. J. Math. Phys. 2006, 13: 340–348. 10.1134/S1061920806030095MathSciNetView ArticleMATHGoogle Scholar
  24. Simsek Y: Theorems on twisted L -function and twisted Bernoulli numbers. Adv. Stud. Contemp. Math. 2005, 11: 205–218.MathSciNetMATHGoogle Scholar
  25. Simsek Y: Special functions related to Dedekind-type sums and their applications. Russ. J. Math. Phys. 2011, 17: 495–508.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Jang; licensee Springer 2012

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.

Advertisement