Ramanujan’s formula for the harmonic number

Abstract

In this paper, we investigate certain asymptotic series used by Hirschhorn to prove an asymptotic expansion of Ramanujan for the nth harmonic number. We give a general form of these series with a recursive formula for its coefficients. By using the result obtained, we present a formula for determining the coefficients of Ramanujan’s asymptotic expansion for the nth harmonic number. We also give a recurrence relation for determining the coefficients aj(r) such thatHn:=∑k=1n1k∼12ln(2m)+γ+112m(∑j=0∞aj(r)mj)1/ras n → ∞, where m=n(n+1)/2 is the nth triangular number and γ is the Euler–Mascheroni constant. In particular, for r=1, we obtain Ramanujan’s expansion for the harmonic number.