Skip to main content

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

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.

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(x1)(xn+1) (n0). The Stirling number of the second kind is defined by

x n = l = 0 n S 2 (n,l) ( x ) l (n0).
(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 (kZ)(see [8–10]).
(5)

For k=1, Li 1 (x)= n = 1 x n n =log(1x). The Stirling number of the first kind is defined by

( x ) n = l = 0 n S 1 (n,l) x l (n0)(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 kZ, 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)(n0).
(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)(n0).

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 n0 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 n0, 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 n1, 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 n0, we have

b n ( k ) (x+y)= l = 0 n ( n l ) b n l ( k ) (x) ( y ) l .

References

  1. 1.

    Kim DS, Kim T, Lee S-H: Poly-Cauchy numbers and polynomials with umbral calculus viewpoint. Int. J. Math. Anal. 2013, 7: 2235-2253.

    Google Scholar 

  2. 2.

    Prabhakar TR, Gupta S: Bernoulli polynomials of the second kind and general order. Indian J. Pure Appl. Math. 1980, 11: 1361-1368.

    MathSciNet  Google Scholar 

  3. 3.

    Roman S, Rota GC: The umbral calculus. Adv. Math. 1978, 27(2):95-188. 10.1016/0001-8708(78)90087-7

    MathSciNet  Article  Google Scholar 

  4. 4.

    Choi J, Kim DS, Kim T, Kim YH: Some arithmetic identities on Bernoulli and Euler numbers arising from the p -adic integrals on Z p . Adv. Stud. Contemp. Math. 2012, 22: 239-247.

    Google Scholar 

  5. 5.

    Ding D, Yang J: Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials. Adv. Stud. Contemp. Math. 2010, 20: 7-21.

    MathSciNet  Google Scholar 

  6. 6.

    Gaboury S, Tremblay R, Fugère B-J: Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials. Proc. Jangjeon Math. Soc. 2014, 17: 115-123.

    MathSciNet  Google Scholar 

  7. 7.

    King D, Lee SJ, Park L-W, Rim S-H: On the twisted weak weight q -Bernoulli polynomials and numbers. Proc. Jangjeon Math. Soc. 2013, 16: 195-201.

    MathSciNet  Google Scholar 

  8. 8.

    Kim DS, Kim T, Lee S-H: A note on poly-Bernoulli polynomials arising from umbral calculus. Adv. Stud. Theor. Phys. 2013, 7(15):731-744.

    Google Scholar 

  9. 9.

    Kim DS, Kim T: Higher-order Frobenius-Euler and poly-Bernoulli mixed-type polynomials. Adv. Differ. Equ. 2013., 2013: Article ID 251

    Google Scholar 

  10. 10.

    Kim DS, Kim T, Lee S-H, Rim S-H: Umbral calculus and Euler polynomials. Ars Comb. 2013, 112: 293-306.

    MathSciNet  Google Scholar 

  11. 11.

    Kim DS, Kim T: Higher-order Cauchy of first kind and poly-Cauchy of the first kind mixed type polynomials. Adv. Stud. Contemp. Math. 2013, 23(4):621-636.

    MathSciNet  Google Scholar 

  12. 12.

    Kim T: q -Bernoulli numbers and polynomials associated with Gaussian binomial coefficients. Russ. J. Math. Phys. 2008, 15: 51-57. 10.1134/S1061920808010068

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kim Y-H, Hwang K-W: Symmetry of power sum and twisted Bernoulli polynomials. Adv. Stud. Contemp. Math. 2009, 18: 127-133.

    MathSciNet  Google Scholar 

  14. 14.

    Ozden H, Cangul IN, Simsek Y: Remarks on q -Bernoulli numbers associated with Daehee numbers. Adv. Stud. Contemp. Math. 2009, 18: 41-48.

    MathSciNet  Google Scholar 

  15. 15.

    Park J-W: New approach to q -Bernoulli polynomials with weight or weak weight. Adv. Stud. Contemp. Math. 2014, 24(1):39-44.

    MathSciNet  Google Scholar 

  16. 16.

    Roman S Pure and Applied Mathematics 111. In The Umbral Calculus. Academic Press, New York; 1984.

    Google Scholar 

  17. 17.

    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  Google Scholar 

  18. 18.

    Srivastava HM, Kim T, Simsek Y: q -Bernoulli numbers and polynomials associated with multiple q -zeta functions and basic L -series. Russ. J. Math. Phys. 2005, 12: 241-278.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Taekyun Kim.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kim, T., Kwon, H.I., Lee, S.H. et al. A note on poly-Bernoulli numbers and polynomials of the second kind. Adv Differ Equ 2014, 219 (2014). https://doi.org/10.1186/1687-1847-2014-219

Download citation

Keywords

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