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
\(\ds \) \(=\) \(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \paren {\harm 1 {x - \dfrac k n} } + \ln n\)

$\blacksquare$


Also see

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