SOME IDENTITIES FOR BERNOULLI NUMBERS OF THE SECOND KIND ARISING FROM A NON-LINEAR DIFFERENTIAL EQUATION

Abstract

Abstract. In this paper, we give explicit and new identities for theBernoulli numbers of the second kind which are derived from a non-lineardifferential equation. 1. IntroductionFor r ∈ N, the Bernoulli polynomials of order r are defined by the generatingfunction to bete t −1 r e xt =te t −1×···×te t −1| {z } r-times (1) e xt= X ∞n=0 B (r)n (x)t n n!, (see [1]-[16]).When x = 0, B (r)n = B (r)n (0) are called the Bernoulli numbers of order r.As is well known, the Bernoulli polynomials of the second kind are given bythe generating function to be(2)tlog(1+t)(1+t) x =X ∞n=0 b n (x)t n n!, (see [3, 5, 7, 14]).Indeed, b n (x) = B (n)n (x +1).When x = 0, b n = b n (0) are called the Bernoulli numbers of the secondkind.The first few Bernoulli numbers of the second kind are b 0 = 1, b 1 = 12 ,b 2 = − 16 , b 3 = 14 , b 4 = − 1930 , b 5 = 94 , .... Received October 8, 2014; Revised May 12, 2015.2010 Mathematics Subject Classification. 05A19,11B68,34A34.Key words and phrases. Bernoulli numbers of second kind, non-linear differentialequation.