• KIM, DAE SAN (Department of Mathematics Sogang University) ;
  • KIM, TAEKYUN (Department of Mathematics Kwangwoon University)
  • Received : 2014.10.08
  • Published : 2015.11.30


In this paper, we give explicit and new identities for the Bernoulli numbers of the second kind which are derived from a non-linear differential equation.


  1. S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 399-406.
  2. A. Bayad and T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 2, 247-253.
  3. L. Comtet, Advanced Combinatorics, Revised and enlarged ed., D. Reidel Publishing Co., Dordrecht, 1974.
  4. K.-W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H. Lee, Some theorems on Bernoulli and Euler numbers, Ars Combin. 109 (2013), 285-297.
  5. H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1988.
  6. D. Kang, J. Jeong, S.-H. Lee, and S.-J. Rim, A note on the Bernoulli polynomials arising from a non-linear differential equation, Proc. Jangjeon Math. Soc. 16 (2013), no. 1, 37-43.
  7. D. S. Kim and T. Kim, Higher-order Cauchy of the first kind and poly-Cauchy of the first kind mixed type polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 4, 621-636.
  8. D. S. Kim, T. Kim, Y.-H. Kim, and D. V. Dolgy, A note on Eulerian polynomials associated with Bernoulli and Euler numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 3, 379-389.
  9. G. Kim, B. Kim, and J. Choi, The DC algorithm for computing sums of powers of consecutive integers and Bernoulli numbers, Adv. Stud. Contemp. Math. (Kyungshang) 17 (2008), no. 2, 137-145.
  10. T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients, Russ. J. Math. Phys. 15 (2008), no. 1, 51-57.
  11. T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, J. Number Theory 132 (2012), no. 12, 2854-2865.
  12. Y.-H. Kim and K.-W. Hwang, Symmetry of power sum and twisted Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 2, 127-133.
  13. H. Ozden, I. N. Cangul, and Y. Simsek, Remarks on q-Bernoulli numbers associated with Daehee numbers, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 1, 41-48.
  14. S. Roman, The umbral calculus, Pure and Applied Mathematics, vol. 111, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1984.
  15. E. Sen, Theorems on Apostol-Euler polynomials of higher order arising from Euler basis, Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 2, 337-345.
  16. Y. Simsek, Generating functions of the twisted Bernoulli numbers and polynomials as- sociated with their interpolation functions, Adv. Stud. Contemp. Math. (Kyungshang) 16 (2008), no. 2, 251-278.

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