Digamma Additive Formula/Corollary
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Corollary to Digamma Additive Formula
Let $n \in \N_{>0}$ where $\N_{>0}$ denotes the non-zero natural numbers.
Then:
- $\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n} = -\paren {n - 1} \gamma - n \ln n$
where:
- $\psi$ is the digamma function
- $\ln$ is the complex natural logarithm.
- $\gamma$ denotes the Euler-Mascheroni constant.
Proof
\(\ds \frac 1 n \sum_{k \mathop = 0}^{n - 1} \map \psi {z + \frac k n} + \ln n\) | \(=\) | \(\ds \map \psi {n z}\) | Digamma Additive Formula | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 0}^{n - 1} \map \psi {z + \frac k n} + n \ln n\) | \(=\) | \(\ds n \map \psi {n z}\) | multiplying both sides by $n$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 0}^{n - 1} \map \psi {z + \frac k n}\) | \(=\) | \(\ds n \map \psi {n z} - n \ln n\) | subtracting $n \ln n$ from both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {z + \frac k n} + \map \psi z\) | \(=\) | \(\ds n \map \psi {n z} - n \ln n\) | reindexing the sum | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {z + \frac k n}\) | \(=\) | \(\ds n \map \psi {n z} - \map \psi z - n \ln n\) | subtracting $\map \psi z$ from both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \lim_{z \mathop \to 0} \paren {\sum_{k \mathop = 1}^{n - 1} \map \psi {z + \frac k n} }\) | \(=\) | \(\ds \lim_{z \mathop \to 0} \paren {n \map \psi {n z} - \map \psi z} - n \ln n\) | taking the limit as $z$ approaches $0$ on both sides | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 1}^{n - 1} \map \psi {\frac k n}\) | \(=\) | \(\ds \lim_{z \mathop \to 0} \paren {n \paren {-\gamma + \sum_{k \mathop = 1}^\infty \paren {\frac 1 k - \frac 1 {n z + k - 1} } } - \paren {-\gamma + \sum_{k \mathop = 1}^\infty \paren {\frac 1 k - \frac 1 {z + k - 1} } } } - n \ln n\) | Reciprocal times Derivative of Gamma Function | ||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {n - 1} \gamma + \lim_{z \mathop \to 0} \paren {n \paren {\sum_{k \mathop = 1}^\infty \paren {\frac 1 k - \frac 1 {n z + k - 1} } } - \paren {\sum_{k \mathop = 1}^\infty \paren {\frac 1 k - \frac 1 {z + k - 1} } } } - n \ln n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {n - 1} \gamma + \lim_{z \mathop \to 0} \paren {\paren {\paren {\frac n 1 - \frac n {n z} } + \paren {\frac n 2 - \frac n {n z + 1} } + \paren {\frac n 3 - \frac n {n z + 2} } + \cdots } - \paren {\paren {\frac 1 1 - \frac 1 z} + \paren {\frac 1 2 - \frac 1 {z + 1} } + \paren {\frac 1 3 - \frac 1 {z + 2} } + \cdots } } - n \ln n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {n - 1} \gamma + \paren {\paren {\frac n 1 - \frac 1 0} + \paren {\frac n 2 - \frac n 1} + \paren {\frac n 3 - \frac n 2} + \cdots} - \paren {\paren {\frac 1 1 - \frac 1 0} + \paren {\frac 1 2 - \frac 1 1} + \paren {\frac 1 3 - \frac 1 2} + \cdots} - n \ln n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\paren {n - 1} \gamma - n \ln n\) |
$\blacksquare$
Sources
- 1985: Bruce C. Berndt: Ramanujan's Notebooks: Part I: Chapter $8$. Analogues of the Gamma Function