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The twisted Daehee numbers and polynomials

Advances in Difference Equations20142014:1

https://doi.org/10.1186/1687-1847-2014-1

  • Received: 30 October 2013
  • Accepted: 6 December 2013
  • Published:

Abstract

We consider the Witt-type formula for the n th twisted Daehee numbers and polynomials and investigate some properties of those numbers and polynomials. In particular, the n th twisted Daehee numbers are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind.

Keywords

  • the n th twisted Daehee numbers and polynomials
  • Bernoulli numbers of the second kind
  • higher-order Bernoulli numbers

1 Introduction

In this paper, we assume that Z p , Q p and C p will, respectively, denote the rings of p-adic integers, the fields of p-adic numbers and the completion of algebraic closure of Q p . The p-adic norm | | p is normalized by | p | p = 1 / p . Let UD [ Z p ] be the space of uniformly differentiable functions on Z p . For f UD [ Z p ] , the p-adic invariant integral on Z p is defined by
I ( f ) Z p f ( x ) d μ 0 ( x ) = lim n 1 p n x = 0 p n 1 f ( x ) ( see [1, 2] ) .
(1)
Let f 1 be the translation of f with f 1 ( x ) = f ( x + 1 ) . Then, by (1), we get
I ( f 1 ) = I ( f ) + f ( 0 ) , where  f ( 0 ) = d f ( x ) d x | x = 0 .
(2)
As is known, the Stirling number of the first kind is defined by
( x ) n = x ( x 1 ) ( x n + 1 ) = l = 0 n S 1 ( n , l ) x l ,
(3)
and the Stirling number of the second kind is given by the generating function to be
( e t 1 ) m = m ! l = m S 2 ( l , m ) t l l ! ( see [3–5] ) .
(4)
For α Z , the Bernoulli polynomials of order α are defined by the generating function to be
( t e t 1 ) α e x t = n = 0 B n ( α ) ( x ) t n n ! ( see [3, 6, 7] ) .
(5)

When x = 0 , B n ( α ) = B n ( α ) ( 0 ) are called the Bernoulli numbers of order α.

For n N , let T p be the p-adic locally constant space defined by
T p = n 1 C p n = lim n C p n ,
where C p n = { ω | ω p n = 1 } is the cyclic group of order p n . It is well known that the twisted Bernoulli polynomials are defined as
t ξ e t 1 e x t = n = 0 B n , ξ ( x ) t n n ! , ξ T p ( see [8] ) ,

and the twisted Bernoulli numbers B n , ξ are defined as B n , ξ = B n , ξ ( 0 ) .

Recently, Kim and Kim introduced the Daehee numbers and polynomials which are given by the generating function to be
( log ( 1 + t ) t ) ( 1 + t ) x = n = 0 D n ( x ) t n n ! ( see [9, 10] ) .
(6)

In the special case, x = 0 , D n = D n ( 0 ) are called the n th Daehee numbers.

In the viewpoint of generalization of the Daehee numbers and polynomials, we consider the n th twisted Daehee polynomials defined by the generating function to be
( log ( 1 + ξ t ) ξ t ) ( 1 + ξ t ) x = n = 0 D n , ξ ( x ) t n n !
(7)

In the special case, x = 0 , D n , ξ = D n , ξ ( 0 ) are called the n th twisted Daehee numbers.

In this paper, we give a p-adic integral representation of the n th twisted Daehee numbers and polynomials, which are called the Witt-type formula for the n th twisted Daehee numbers and polynomials. We can derive some interesting properties related to the n th twisted Daehee numbers and polynomials. For this idea, we are indebted to papers [9, 10].

2 Witt-type formula for the n th twisted Daehee numbers and polynomials

First, we consider the following integral representation associated with falling factorial sequences:
Z p ( x ) n d μ 0 ( x ) , where  n Z + = N { 0 } ( see [10] ) .
(8)
By (8), we get
n = 0 ξ n Z p ( x ) n d μ 0 ( x ) t n n ! = Z p n = 0 ξ n ( x n ) t n d μ 0 ( x ) = Z p ( 1 + ξ t ) x d μ 0 ( x ) ,
(9)

where t C p with | t | p < 1 p 1 .

For t C p with | t | p < p 1 p 1 , let us take f ( x ) = ( 1 + ξ t ) x . Then, from (2), we have
Z p ( 1 + ξ t ) x d μ 0 ( x ) = log ( 1 + ξ t ) ξ t .
(10)
By (9) and (10), we see that
n = 0 D n , ξ t n n ! = log ( 1 + ξ t ) ξ t = Z p ( 1 + ξ t ) x d μ 0 ( x ) = n = 0 ξ n Z p ( x ) n d μ 0 ( x ) t n n ! .
(11)

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

Theorem 1 For n 0 , we have
ξ n Z p ( x ) n d μ 0 ( x ) = D n , ξ .
For n Z , it is known that
( log ( 1 + t ) t ) n ( 1 + t ) x 1 = k = 0 B k ( k n + 1 ) ( x ) t k k ! ( see [3–5] ) .
(12)
Thus, replacing t by e ξ t 1 in (12), we get
D k , ξ = ξ n Z p ( x ) k d μ 0 ( x ) = ξ n B k ( k + 2 ) ( 1 ) ( k 0 ) ,
(13)

where B k ( n ) ( x ) are the Bernoulli polynomials of order n.

In the special case, x = 0 , B k ( n ) = B k ( n ) ( 0 ) are called the n th Bernoulli numbers of order n.

From (11), we note that
( 1 + ξ t ) x Z p ( 1 + ξ t ) y d μ 0 ( y ) = ( log ( 1 + ξ t ) ξ t ) ( 1 + ξ t ) x = n = 0 D n , ξ ( x ) t n n ! .
(14)
Thus, by (14), we get
ξ n Z p ( x + y ) n d μ 0 ( y ) = D n , ξ ( x ) ( n 0 ) ,
(15)
and, from (12), we have
D n , ξ ( x ) = ξ n B n ( n + 2 ) ( x + 1 ) .
(16)

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

Theorem 2 For n 0 , we have
D n , ξ ( x ) = ξ n Z p ( x + y ) n d μ 0 ( y )
and
D n , ξ ( x ) = ξ n B n ( n + 2 ) ( x + 1 ) .
By Theorem 1, we easily see that
D n , ξ = ξ n l = 0 n S 1 ( n , l ) B l ,
(17)

where B l are the ordinary Bernoulli numbers.

From Theorem 2, we have
D n , ξ ( x ) = ξ n Z p ( x + y ) n d μ 0 ( y ) = ξ n l = 0 n S 1 ( n , l ) B l ( x ) ,
(18)
where B l ( x ) are the Bernoulli polynomials defined by a generating function to be
t e t 1 e x t = n = 0 B n ( x ) t n n ! .

Therefore, by (17) and (18), we obtain the following corollary.

Corollary 3 For n 0 , we have
D n , ξ ( x ) = ξ n l = 0 n S 1 ( n , l ) B l ( x ) .
In (11), we have
log ( 1 + ξ t ) ξ t ( 1 + ξ t ) x = n = 0 D n , ξ ( x ) t n n ! .
(19)
Replacing t by e t 1 ξ , we put
n = 0 D n , ξ ( x ) 1 n ! ( e t 1 ξ ) n = log ( 1 + ξ ( e t 1 ξ ) ) ξ ( e t 1 ξ ) ( 1 + ξ ( e t 1 ξ ) ) x = t ξ e t 1 ( ξ e t ) x = ξ x t ξ e t 1 e t x = ξ x n = 0 B n , ξ ( x ) t n n ! .
(20)
Therefore, we have
n = 0 B n , ξ ( x ) t n n ! = n = 0 D n , ξ ( x ) 1 n ! ( e t 1 ξ ) n = n = 0 D n , ξ ( x ) 1 n ! ξ n n ! m = n S 2 ( m , n ) t m m ! = m = 0 n = 0 m D n , ξ ( x ) ξ n S 2 ( m , n ) ,
(21)

where S 2 ( m , n ) is the Stirling number of the second kind.

Hence,
ξ x B n , ξ ( x ) = n = 0 m D n , ξ ( x ) ξ n S 2 ( m , n ) .
(22)
Therefore, we have
B m , ξ ( x ) = n = 0 m D n , ξ ( x ) ξ n x S 2 ( m , n ) .
(23)
In particular,
B m , ξ = n = 0 m D n , ξ ξ n S 2 ( m , n ) .
(24)

Therefore, by (20) and (23), we obtain the following theorem.

Theorem 4 For m 0 , we have
B m , ξ ( x ) = n = 0 m ξ n x D n , ξ ( x ) S 2 ( m , n ) .
In particular,
B m , ξ = n = 0 m ξ n D n , ξ S 2 ( m , n ) .
Remark For m 0 , by (18), we have
ξ n Z p ( x + y ) m d μ 0 ( y ) = ξ n n = 0 m D n ( x ) S 2 ( m , n ) .
For n Z n 0 , the rising factorial sequence is defined by
x ( n ) = x ( x + 1 ) ( x + n 1 ) .
(25)
Let us define the n th twisted Daehee numbers of the second kind as follows:
D ˆ n , ξ = ξ n Z p ( x ) n d μ 0 ( x ) ( n Z n 0 ) .
(26)
By (26), we get
x ( n ) = ( 1 ) n ( x ) n = l = 0 n S 1 ( n , l ) ( 1 ) n l x l .
(27)
From (26) and (27), we have
D ˆ n , ξ = ξ n Z p ( x ) n d μ 0 ( x ) = ξ n Z p x ( n ) ( 1 ) n d μ 0 ( x ) = ξ n l = 0 n S 1 ( n , l ) ( 1 ) l B l .
(28)

Therefore, by (28), we obtain the following theorem.

Theorem 5 For n 0 , we have
D ˆ n , ξ = ξ n l = 0 n S 1 ( n , l ) ( 1 ) l B l .
Let us consider the generating function of the n th twisted Daehee numbers of the second kind as follows:
n = 0 D ˆ n , ξ t n n ! = n = 0 ξ n Z p ( x ) n d μ 0 ( x ) t n n ! = Z p n = 0 ξ n ( x n ) t n d μ 0 ( x ) = Z p ( 1 + ξ t ) x d μ 0 ( x ) .
(29)
From (2), we can derive the following equation:
Z p ( 1 + ξ t ) x d μ 0 ( x ) = ( 1 + ξ t ) log ( 1 + ξ t ) ξ t ,
(30)

where | t | p < p 1 p .

By (29) and (30), we get
1 ξ t ( 1 + ξ t ) log ( 1 + ξ t ) = Z p ( 1 + ξ t ) x d μ 0 ( x ) = n = 0 D ˆ n , ξ t n n ! .
(31)
Let us consider the n th twisted Daehee polynomials of the second kind as follows:
( 1 + ξ t ) log ( 1 + ξ t ) ξ t 1 ( 1 + ξ t ) x = n = 0 D ˆ n , ξ ( x ) t n n ! .
(32)
Then, by (32), we get
Z p ( 1 + ξ t ) x y d μ 0 ( y ) = n = 0 D ˆ n , ξ ( x ) t n n ! .
(33)
From (33), we get
D ˆ n , ξ ( x ) = ξ n Z p ( x y ) n d μ 0 ( y ) ( n 0 ) = ξ n l = 0 n ( 1 ) l S 1 ( n , l ) B l ( x ) .
(34)

Therefore, by (34), we obtain the following theorem.

Theorem 6 For n 0 , we have
D ˆ n , ξ ( x ) = ξ n Z p ( x y ) n d μ 0 ( y ) = ξ n l = 0 n ( 1 ) l S 1 ( n , l ) B l ( x ) .
From (32) and (33), we have
log ( 1 + ξ t ) ξ t ( 1 + ξ t ) 1 x = n = 0 D ˆ n , ξ ( x ) t n n ! .
(35)
Replacing t by e t 1 ξ , we get
n = 0 D ˆ n , ξ ( x ) 1 n ! ( e t 1 ξ ) n = log ( 1 + ξ ( e t 1 ξ ) ) ξ ( e t 1 ξ ) ( 1 + ξ ( e t 1 ξ ) ) 1 x = t ξ e t 1 ( ξ e t ) 1 x = ξ 1 x t ξ e t 1 e t ( 1 x ) = ξ 1 x n = 0 B n , ξ ( 1 x ) t n n ! .
(36)
Therefore, we have
ξ 1 x m = 0 B m , ξ ( 1 x ) t n n ! = n = 0 D ˆ n , ξ ( x ) ( e t 1 ξ ) n n ! = n = 0 D ˆ n , ξ ( x ) 1 n ! ξ n n ! m = n S 2 ( m , n ) t m m ! = m = 0 ( n = 0 m D ˆ n , ξ ( x ) ξ n S 2 ( m , n ) ) t m m ! .
(37)
Hence,
ξ 1 x B n , ξ ( 1 x ) = n = 0 m D ˆ n , ξ ( x ) ξ n S 2 ( m , n ) .
(38)
Therefore, we have
B m , ξ ( 1 x ) = n = 0 m D ˆ n , ξ ( x ) ξ n + x 1 S 2 ( m , n ) .
(39)

Therefore, by (37) and (38), we obtain the following theorem.

Theorem 7 For m 0 , we have
B m , ξ ( 1 x ) = n = 0 m ξ m + x 1 D ˆ n , ξ ( x ) S 2 ( m , n ) .
From Theorem 1 and (26), we have
( 1 ) n D n , ξ n ! = ( 1 ) n ξ n Z p ( x n ) d μ 0 ( x ) = ξ n Z p ( x + n 1 n ) d μ 0 ( x ) = ξ n m = 0 n ( n 1 n m ) Z p ( x m ) d μ 0 ( x ) = m = 0 n ( n 1 n m ) ξ n m D ˆ m , ξ m ! = m = 1 n ( n 1 m 1 ) ξ n m D ˆ m , ξ m !
(40)
and
( 1 ) n D ˆ n , ξ n ! = ( 1 ) n ξ n Z p ( x n ) d μ 0 ( x ) = ξ n Z p ( x + n 1 n ) d μ 0 ( x ) = ξ n m = 0 n ( n 1 n m ) 0 1 ( x m ) d μ 0 ( x ) = m = 0 n ( n 1 m 1 ) ξ n m D m , ξ m ! = m = 1 n ( n 1 m 1 ) ξ n m D m , ξ m ! .
(41)

Therefore, by (40) and (41), we obtain the following theorem.

Theorem 8 For n N , we have
( 1 ) n D n , ξ n ! = m = 1 n ( n 1 m 1 ) ξ n m D ˆ m , ξ m !
and
( 1 ) n D ˆ n , ξ n ! = m = 1 n ( n 1 m 1 ) ξ n m D m , ξ m ! .

Declarations

Acknowledgements

The authors are grateful for the valuable comments and suggestions of the referees.

Authors’ Affiliations

(1)
Department of Mathematics Education, Kyungpook National University, Taegu, 702-701, Republic of Korea
(2)
Department of Mathematics, Kyungpook National University, Taegu, 702-701, Republic of Korea

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Copyright

© Park et al.; licensee Springer. 2014

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.

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