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

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 $$H_n=\frac{1}{m}\sum _{k=1}^n\frac{(-1{)}^{k+1}}{k}\left(\genfrac{}{}{0px}{}{mk}{k}\right)\left(\genfrac{}{}{0px}{}{n+(m-1)k}{n-k}\right)\mathrm{.}$$ 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 $\text{QQ}[[x]]$. The proof reduces, via a substitution $u=x\mathrm{/}(1-x{)}^m$, to two formal-power-series identities: a Lagrange–Bürmann evaluation of ${\sum }_{k\ge 1}\left(\genfrac{}{}{0px}{}{mk}{k}\right)u^k\mathrm{/}k$, 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 $m\mathrm{\ne }0$.