Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion

Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: H n ∼ γ + log n − ∑ k = 1 ∞ B k k · n k , where B k 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: H n ∼ γ + c 0 ( h ) log ( q + h ) − ∑ k = 1 ∞ c k ( h ) k · ( q + h ) k , where q = n ( n + 1 ) is the n th pronic number, twice the n th triangular number, γ is the Euler–Mascheroni constant, and c k ( x ) = ∑ j = 0 k k j c j x k − j , with c k is the negative of the median Bernoulli numbers. Then, 2 c n = ∑ k = 0 n n k B n + k , where B n is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the n th harmonic number. For example, H n ∼ γ + 1 2 log ( q + 1 3 ) − 1 180 ( q + 1 3 ) 2 ∑ j = 0 ∞ b j ( r ) ( q + 1 3 ) j 1 / r as n approaches infinity, where b j ( r ) can be determined.