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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: 5 August 2014

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, Republic of Korea
(2)
Division of General Education, Kwangwoon University, Seoul, Republic of Korea
(3)
Department of Applied Mathematics, Pukyong National University, Pusan, 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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