A Proof of Bala's General-$m$ Representation of the Harmonic Numbers

TL;DR

This paper proves Bala's conjectured generalization of harmonic number identities for all nonzero integers m, using formal power series and combinatorial identities.

Contribution

It provides a formal proof of Bala's conjecture, extending the classical cases to all nonzero integer m.

Findings

Proves Bala's identity for all nonzero integer m.

Reduces the proof to formal power series identities.

Extends the result to complex m ≠ 0.

Abstract

For every nonzero integer $m$ and every integer $n \ge 1$, the $n$\textsuperscript{th} harmonic number $H_n = 1 + \tfrac12 + \dots + \tfrac1n$ satisfies the identity \[ H_n \;=\; \frac{1}{m}\,\sum_{k=1}^{n} \frac{(-1)^{k+1}}{k}\, \binom{m k}{k}\binom{n + (m-1)k}{n - k}. \] The cases $m = 1$ and $m = 2$ are classical; for general nonzero integer $m$ the identity was conjectured by P.~Bala in the OEIS entry A001008 in 2022 and remained open. We prove it here, working throughout in $\QQ[[x]]$. The proof reduces, via a substitution $u = x/(1-x)^m$, to two formal-power-series identities: a Lagrange--B\"urmann evaluation of $\sum_{k\ge1} \binom{mk}{k} u^k / k$, and the fixed-point fact that under that substitution the unique solution $v(u)$ of $v = u(1-v)^{m}$ is $v = x$. The argument extends verbatim to arbitrary complex $m \ne 0$.