Closed and asymptotic formulae for harmonic and quadratic harmonic sums

TL;DR

This paper develops a framework to obtain closed-form finite-n and asymptotic expressions for quadratic harmonic sums by introducing the sum families $G_{n,p,q}^{r,s,m}$ and $V_{n,p,q}^{r,s,m}$ and deriving a general recursive approach to reduce these sums to tractable building blocks such as harmonic numbers and zeta values. It provides explicit recursive formulæ for computing these sums, and applies the method to a range of cases with $r,s,p,q\in\{1,2\}$ and $m\in\{1,2\}$, yielding numerous closed forms and limits. The work includes an extensive Appendix of harmonic-sum identities, numerical verification, and a publicly available Mathematica implementation. The results advance practical evaluation of sums arising in probabilistic and applied contexts, clarifying the structure of harmonic and quadratic-harmonic sums and supplying concrete closed-form expressions and limits in terms of $\zeta$ values and powers of $\pi$.

Abstract

We present here a large collection of harmonic and quadratic harmonic sums, that can be useful in applied questions, e.g., probabilistic ones. We find closed-form formulae, that we were not able to locate in the literature.

Theorems & Definitions (128)

• Theorem 3.1
• Theorem 3.2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8