Certain Mathematical Constants Associated with Harmonic Numbers and Higher-Dimensional Harmonic Sums

The Euler–Mascheroni constant γ, defined as the limiting difference between the harmonic numbers Hn and logn, has long been studied and appears in diverse areas of number theory, analysis, and special functions. In this paper, we establish a unified formula for (k+1)-fold harmonic sums expressed in terms of harmonic numbers. Several particular cases are examined in detail, and their asymptotic expansions are derived, leading to the identification of both classical and additional limiting constants. These results place higher-order harmonic sums within a common analytic framework and clarify the structure of their normalized limits. The broader mathematical significance of the additional constants arising from this approach remains to be determined and may warrant further investigation.