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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: 2 January 2014

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, Republic of Korea
(2)
Department of Mathematics, Kyungpook National University, Taegu, 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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