Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion

Abstract

Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers. In this paper, we rewrite Ramanujan’s harmonic number expansion into a similar form of Euler’s asymptotic expansion as n approaches infinity: Hn∼γ+c0(h)log(q+h)−∑k=1∞ck(h)k·(q+h)k, where q=n(n+1) is the nth pronic number, twice the nth triangular number, γ is the Euler–Mascheroni constant, and ck(x)=∑j=0kkjcjxk−j, with ck is the negative of the median Bernoulli numbers. Then, 2cn=∑k=0nnkBn+k, where Bn is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the nth harmonic number. For example, Hn∼γ+12log(q+13)−1180(q+13)2∑j=0∞bj(r)(q+13)j1/r as n approaches infinity, where bj(r) can be determined.