Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion - Kwang-Wu Chen (Mathematics, 2022)
Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: <semantics>Hn∼γ+logn−<msubsup>∑k=1∞</msubsup>Bkk·nk</semantics>, where <semantics>Bk</semantics> 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: <semantics>Hn∼γ+c0(h)log(q+h)−<msubsup>∑k=1∞</msubsup>ck(h)k·(q+h)k</semantics>, where <semantics>q=n(n+1)</semantics> is the nth pronic number, twice the nth triangular number, <semantics>γ</semantics> is the Euler–Mascheroni constant, and <semantics>ck(x)=<msubsup>∑j=0k</msubsup><mfenced separators="" open="(" close=")">kj</mfenced>cjxk−j</semantics>, with <semantics>ck</semantics> is the negative of the median Bernoulli numbers. Then, <semantics>2cn=<msubsup>∑k=0n</msubsup><mfenced separators="" open="(" close=")">nk</mfenced>Bn+k</semantics>, where <semantics>Bn</semantics> is the Bernoulli number. By using the result obtained, we present two general Ramanujan’s asymptotic expansions for the nth harmonic number. For example, <semantics>Hn∼γ+12log(q+13)−1180(q+13)2<mfenced separators="" open="(" close=")"><msubsup>∑j=0∞</msubsup>bj(r)(q+13)j</mfenced>1/r</semantics> as n approaches infinity, where <semantics>bj(r)</semantics> can be determined.