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

Abstract

For every nonzero integer $m$ and every integer $n\ge 1$, the $n$\textsuperscript{th} harmonic number $H_n=1+\frac{1}{2}+\cdots +\frac{1}{n}$ 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--Bürmann evaluation of ${\sum }_{k\ge 1}\left(\genfrac{}{}{0ex}{}{mk}{k}\right)u^k/k$, 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 $m\ne 0$.

13 pages, v2: added Appendices A (verify_bala.py) and B (check_proof.py) inlining the verification code in full; Section 7 updated to cross-reference. Math content of Sections 1--6 unchanged