A Proof of Bala's General-$m$ Representation of the Harmonic Numbers
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914523308359680 |
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| author | Niu, Tong |
| author_facet | Niu, Tong |
| contents | 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--Bürmann 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 $m \ne 0$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23206 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Proof of Bala's General-$m$ Representation of the Harmonic Numbers Niu, Tong Number Theory Discrete Mathematics Combinatorics 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--Bürmann 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 $m \ne 0$. |
| title | A Proof of Bala's General-$m$ Representation of the Harmonic Numbers |
| topic | Number Theory Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2604.23206 |