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A note on poly-Bernoulli numbers and polynomials of the second kind

  • 1Email author,
  • 1,
  • 2 and
  • 3
Advances in Difference Equations20142014:219

https://doi.org/10.1186/1687-1847-2014-219

  • Received: 23 June 2014
  • Accepted: 24 July 2014
  • Published:

Abstract

In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulas for calculating the poly-Bernoulli numbers of the second kind and the Stirling numbers of the second kind.

Keywords

  • Bernoulli polynomials of the second kind
  • poly-Bernoulli numbers and polynomials
  • Stirling number of the second kind

1 Introduction

As is well known, the Bernoulli polynomials of the second kind are defined by the generating function to be
t log ( 1 + t ) ( 1 + t ) x = n = 0 b n ( x ) t n n ! ( see [1–3] ) .
(1)

When x = 0 , b n = b n ( 0 ) are called the Bernoulli numbers of the second kind. The first few Bernoulli numbers b n of the second kind are b 0 = 1 , b 1 = 1 / 2 , b 2 = 1 / 12 , b 3 = 1 / 24 , b 4 = 19 / 720 , b 5 = 3 / 160 ,  .

From (1), we have
b n ( x ) = l = 0 n ( n l ) b l ( x ) n l ,
(2)
where ( x ) n = x ( x 1 ) ( x n + 1 ) ( n 0 ). The Stirling number of the second kind is defined by
x n = l = 0 n S 2 ( n , l ) ( x ) l ( n 0 ) .
(3)
The ordinary Bernoulli polynomials are given by
t e t 1 e x t = n = 0 B n ( x ) t n n ! ( see [1–18] ) .
(4)

When x = 0 , B n = B n ( 0 ) are called Bernoulli numbers.

It is well known that the classical poly-logarithmic function Li k ( x ) is given by
Li k ( x ) = n = 1 x n n k ( k Z ) ( see [8–10] ) .
(5)
For k = 1 , Li 1 ( x ) = n = 1 x n n = log ( 1 x ) . The Stirling number of the first kind is defined by
( x ) n = l = 0 n S 1 ( n , l ) x l ( n 0 ) ( see [16] ) .
(6)

In this paper, we consider the poly-Bernoulli numbers and polynomials of the second kind and presents new and explicit formulas for calculating the poly-Bernoulli number and polynomial and the Stirling number of the second kind.

2 Poly-Bernoulli numbers and polynomials of the second kind

For k Z , we consider the poly-Bernoulli polynomials b n ( k ) ( x ) of the second kind:
Li k ( 1 e t ) log ( 1 + t ) ( 1 + t ) x = n = 0 b n ( k ) ( x ) t n n ! .
(7)

When x = 0 , b n ( k ) = b n ( k ) ( 0 ) are called the poly-Bernoulli numbers of the second kind.

Indeed, for k = 1 , we have
Li k ( 1 e t ) log ( 1 + t ) ( 1 + t ) x = t log ( 1 + t ) ( 1 + t ) x = n = 0 b n ( x ) t n n ! .
(8)
By (7) and (8), we get
b n ( 1 ) ( x ) = b n ( x ) ( n 0 ) .
(9)
It is well known that
t ( 1 + t ) x 1 log ( 1 + t ) = n = 0 B n ( n ) ( x ) t n n ! ,
(10)
where B n ( α ) ( x ) are the Bernoulli polynomials of order α which are given by the generating function to be
( t e t 1 ) α e x t = n = 0 B n ( α ) ( x ) t n n ! ( see [1–18] ) .
By (1) and (10), we get
b n ( x ) = B n ( n ) ( x + 1 ) ( n 0 ) .
Now, we observe that
(11)
Thus, by (11), we get
n = 0 b n ( 2 ) ( x ) t n n ! = ( 1 + t ) x log ( 1 + t ) 0 t x e x 1 d x = ( 1 + t ) x log ( 1 + t ) l = 0 B l l ! 0 t x l d x = ( t log ( 1 + t ) ) ( 1 + t ) x l = 0 B l ( l + 1 ) t l l ! = n = 0 { l = 0 n ( n l ) B l b n l ( x ) l + 1 } t n n ! .
(12)

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

Theorem 2.1 For n 0 we have
b n ( 2 ) ( x ) = l = 0 n ( n l ) B l b n l ( x ) l + 1 .
From (11), we have
n = 0 b n ( k ) ( x ) t n n ! = Li k ( 1 e t ) log ( 1 + t ) ( 1 + t ) x = t log ( 1 + t ) Li k ( 1 e t ) t ( 1 + t ) x .
(13)
We observe that
1 t Li k ( 1 e t ) = 1 t n = 1 1 n k ( 1 e t ) n = 1 t n = 1 ( 1 ) n n k n ! l = n S 2 ( l , n ) ( t ) l l ! = 1 t l = 1 n = 1 l ( 1 ) n + l n k n ! S 2 ( l , n ) t l l ! = l = 0 n = 1 l + 1 ( 1 ) n + l + 1 n k n ! S 2 ( l + 1 , n ) l + 1 t l l ! .
(14)
Thus, by (10) and (14), we get
n = 0 b n ( k ) ( x ) t n n ! = ( m = 0 b m ( x ) t m m ! ) { l = 0 ( p = 1 l + 1 ( 1 ) p + l + 1 p k p ! S 2 ( l + 1 , p ) l + 1 ) t l l ! } = n = 0 { l = 0 n ( n l ) ( p = 1 l + 1 ( 1 ) p + l + 1 p ! p k S 2 ( l + 1 , p ) l + 1 ) b n l ( x ) } t n n ! .
(15)

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

Theorem 2.2 For n 0 , we have
b n ( k ) ( x ) = l = 0 n ( n l ) ( p = 1 l + 1 ( 1 ) p + l + 1 p k p ! S 2 ( l + 1 , p ) l + 1 ) b n l ( x ) .
By (7), we get
n = 0 ( b n ( k ) ( x + 1 ) b n ( k ) ( x ) ) t n n ! = Li k ( 1 e t ) log ( 1 + t ) ( 1 + t ) x + 1 Li k ( 1 e t ) log ( 1 + t ) ( 1 + t ) x = t Li k ( 1 e t ) log ( 1 + t ) ( 1 + t ) x = ( t log ( 1 + t ) ( 1 + t ) x ) Li k ( 1 e t ) = ( l = 0 b l ( x ) l ! t l ) { p = 1 ( m = 1 p ( 1 ) m + p m ! m k S 2 ( p , m ) ) } t p p !
(16)
= n = 1 ( p = 1 n m = 1 p ( 1 ) m + p m k m ! S 2 ( p , m ) b n p ( x ) n ! ( n p ) ! p ! ) t n n ! = n = 1 { p = 1 n m = 1 p ( n p ) ( 1 ) m + p m ! m k S 2 ( p , m ) b n p ( x ) } t n n ! .
(17)

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

Theorem 2.3 For n 1 , we have
b n ( k ) ( x + 1 ) b n ( k ) ( x ) = p = 1 n m = 1 p ( n p ) ( 1 ) m + p m ! m k S 2 ( p , m ) b n p ( x ) .
(18)
From (13), we have
n = 0 b n ( k ) ( x + y ) t n n ! = ( Li k ( 1 e t ) log ( 1 + t ) ) k ( 1 + t ) x + y = ( Li k ( 1 e t ) log ( 1 + t ) ) k ( 1 + t ) x ( 1 + t ) y = ( l = 0 b l ( k ) ( x ) t l l ! ) ( m = 0 ( y ) m t m m ! ) = n = 0 ( l = 0 n ( y ) l b n l ( k ) ( x ) n ! ( n l ) ! l ! ) t n n ! = n = 0 ( l = 0 n ( n l ) b n l ( k ) ( x ) ( y ) l ) t n n ! .
(19)

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

Theorem 2.4 For n 0 , we have
b n ( k ) ( x + y ) = l = 0 n ( n l ) b n l ( k ) ( x ) ( y ) l .

Declarations

Acknowledgements

The present research has been conducted by the Research Grant of Kwangwoon University in 2014.

Authors’ Affiliations

(1)
Department of Mathematics, Kwangwoon University, Seoul, 139-701, Republic of Korea
(2)
Division of General Education, Kwangwoon University, Seoul, 139-701, Republic of Korea
(3)
Department of Applied Mathematics, Pukyong National University, Pusan, 698-737, Republic of Korea

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Copyright

© Kim 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/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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