General Harmonic Number Additive Formula
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Theorem
Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the non-zero natural numbers.
Let $x \in \C \setminus \Z_{< 0}$
Then:
- $\ds \harm 1 {n x} = \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {x - \dfrac k n } } + \ln n$
where:
- $\map {H^{\paren r} } x$ denotes the general harmonic number of order $1$ evaluated at $x$.
- $\ln$ is the complex natural logarithm.
Corollary
Let $n \in \N_{>0}$ be a non-zero natural number.
Then:
- $\ds \sum_{k \mathop = 1}^{n - 1} \paren {\harm 1 {-\dfrac k n } } = -n \ln n$
where:
- $\harm 1 x$ denotes the general harmonic number of order $1$ evaluated at $x$.
- $\ln$ is the complex natural logarithm.
Proof
\(\ds \map \psi {n z}\) | \(=\) | \(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \map \psi {z + \frac k n} + \ln n\) | Digamma Additive Formula | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {-\gamma + \harm 1 {n z - 1} }\) | \(=\) | \(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {-\gamma + \harm 1 {z + \dfrac k n - 1} } + \ln n\) | Digamma Function in terms of General Harmonic Number: $\map \psi {z + 1} = -\gamma + \harm 1 z$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 n \paren {-n \gamma} + \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {z + \dfrac k n - 1} } + \ln n\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \harm 1 {n z - 1}\) | \(=\) | \(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {z + \dfrac k n - 1} } + \ln n\) | adding $\gamma$ from both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \harm 1 {n x}\) | \(=\) | \(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {x + \dfrac 1 n + \dfrac k n - 1} } + \ln n\) | let $\paren {n z - 1} \to n x$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 n \paren {\harm 1 {x - \dfrac {\paren {n - 1} } n} + \harm 1 {x - \dfrac {\paren {n - 2} } n } + \cdots + \harm 1 {x - \dfrac {\paren {n - n} } n} } + \ln n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 n \paren {\harm 1 {x - \dfrac {\paren {n - n} } n} + \harm 1 {x - \dfrac {\paren {n - \paren {n - 1} } } n } + \cdots + \harm 1 {x - \dfrac {\paren {n - 1} } n} } + \ln n\) | reversing the order of the sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {x - \dfrac k n} } + \ln n\) |
$\blacksquare$