Harmonic Number Expansions of the Ramanujan Type

Abstract

In this paper, we establish three (general) asymptotic expansions of the Ramanujan type for the harmonic numbers, and give the corresponding recurrences of the coefficient sequence or parameter sequences in these expansions. We also present two explicit expressions for the coefficient sequence of the first expansion by the methods of generating functions and summation transformations. It can be found that the first expansion includes the classical Ramanujan formula, the DeTemple–Wang formula and the Chen–Mortici–Villarino formula as special cases, and the third one includes the refinement of Lodge’s approximation as a special case. Moreover, the second and third expansions are lacunary and contain only even power terms or odd power terms. By these expansions, we give unified approaches to dealing with asymptotic expansions of the Ramanujan type for the harmonic numbers.