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

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.

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(xlogq) 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 =1andq ( 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 ) andq ( 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)=(q1) 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)=(q1)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 +(1q) 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 +(1q) 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 +(1q) 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

ξq 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

ξq ( 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 ) (1x)= ( 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

References

  1. Bayad A: Modular properties of elliptic Bernoulli and Euler functions. Adv. Stud. Contemp. Math. 2010, 20: 389–401.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google Scholar 

  3. Carlitz L: Expansion of q -Bernoulli numbers and polynomials. Duke Math. J. 1958, 25: 355–364. 10.1215/S0012-7094-58-02532-8

    Article  MathSciNet  MATH  Google Scholar 

  4. Carlitz L: q -Bernoulli numbers and polynomials. Duke Math. J. 1948, 15: 987–1000. 10.1215/S0012-7094-48-01588-9

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google Scholar 

  7. Hegazi AS, Mansour M: A note on q -Bernoulli numbers and polynomials. J. Nonlinear Math. Phys. 2006, 13: 9–18.

    Article  MathSciNet  MATH  Google 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 163217

    Google Scholar 

  9. Jang LC: A study on the distribution of twisted q -Genocchi polynomials. Adv. Stud. Contemp. Math. 2009, 19: 181–189.

    MathSciNet  MATH  Google Scholar 

  10. Kim T, Choi J: On the q -Bernoulli numbers and polynomials with weight α . Abstr. Appl. Anal. 2011., 2011: Article ID 392025

    Google Scholar 

  11. Kim T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 2008, 15: 51–57.

    Article  MathSciNet  MATH  Google 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/S1061920809040037

    Article  MathSciNet  MATH  Google Scholar 

  13. Kim T: Barnes-type multiple q -zeta functions and q -Euler polynomials. J. Phys. A, Math. Theor. 2010., 43: Article ID 255201

    Google 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.1999.2373

    Article  MathSciNet  MATH  Google Scholar 

  15. Kim T: On the weighted q -Bernoulli numbers and polynomials. Adv. Stud. Contemp. Math. 1999, 21(2):207–215.

    MathSciNet  MATH  Google Scholar 

  16. Kupershmidt BA: Reflection symmetries of q -Bernoulli polynomials. J. Nonlinear Math. Phys. 2005, 12: 412–422. 10.2991/jnmp.2005.12.s1.34

    Article  MathSciNet  Google 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.

    MathSciNet  MATH  Google 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 351419

    Google 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 579509

    Google 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.

    MathSciNet  MATH  Google Scholar 

  21. Ryoo CS: A note on the weighted q -Euler numbers and polynomials. Adv. Stud. Contemp. Math. 2011, 21: 47–54.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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/S1061920806030095

    Article  MathSciNet  MATH  Google Scholar 

  24. Simsek Y: Theorems on twisted L -function and twisted Bernoulli numbers. Adv. Stud. Contemp. Math. 2005, 11: 205–218.

    MathSciNet  MATH  Google Scholar 

  25. Simsek Y: Special functions related to Dedekind-type sums and their applications. Russ. J. Math. Phys. 2011, 17: 495–508.

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

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

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Correspondence to Lee-Chae Jang.

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Jang, LC. Some identities on the twisted q-Bernoulli numbers and polynomials with weight α. J Inequal Appl 2012, 176 (2012). https://doi.org/10.1186/1029-242X-2012-176

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