Sum of Digamma of n plus 2 over n plus 2 squared
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Theorem
| \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \psi {n + 2} } {\paren {n + 2}^2 }\) | \(=\) | \(\ds \dfrac {\map \psi 2} {2^2} + \dfrac {\map \psi 3} {3^2} + \dfrac {\map \psi 4} {4^2} + \dfrac {\map \psi 5} {5^2} + \cdots\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \gamma \cdot \paren {1 - \map \zeta 2} + \map \zeta 3\) |
where:
- $\map \psi n$ is the digamma function of $n$
- $\map \zeta 2$ is the Riemann $\zeta$ function of $2$
- $\map \zeta 3$ is Apéry's constant: the Riemann $\zeta$ function of $3$.
Note the use of the $\cdot$ notation for product, or you wonder what the $\gamma$ function is.
Proof
| \(\ds \sum_{n \mathop = 0}^\infty \dfrac {\map \psi {n + 2} } {\paren {n + 2}^2}\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {\map \psi n} {n^2} - \map \psi 1\) | $n \to n - 2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {\paren {\map \psi {n + 1} - \dfrac 1 n} } {n^2} - \map \psi 1\) | Recurrence Relation for Digamma Function: $\map \psi n = \map \psi {n + 1} - \dfrac 1 n$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {\map \psi {n + 1} } {n^2} - \sum_{n \mathop = 1}^\infty \dfrac 1 {n^3} - \map \psi 1\) | Linear Combination of Convergent Series | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {\map \psi {n + 1} } {n^2} - \map \zeta 3 + \gamma\) | Digamma Function of One and Definition of Apéry's Constant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {-\gamma + \ds \int_{\to 0}^{\to 1} \paren {\dfrac {1 - t^n} {1 - t} } \rd t} {n^2} - \map \zeta 3 + \gamma\) | Reciprocal times Derivative of Gamma Function: Corollary: $\ds \map \psi {n + 1} = -\gamma + \int_{\to 0}^{\to 1} \paren {\dfrac {1 - t^n} {1 - t} } \rd t$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {\ds \int_{\to 0}^{\to 1} \paren {\dfrac {1 - t^n} {1 - t} } \rd t} {n^2} - \gamma \cdot \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2} - \map \zeta 3 +\gamma\) | Linear Combination of Convergent Series | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \dfrac {\ds \int_{\to 0}^{\to 1} \paren {\dfrac {1 - t^n} {1 - t} } \rd t} {n^2} + \gamma \cdot \paren {1 - \map \zeta 2} - \map \zeta 3\) | Definition of Riemann Zeta Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \dfrac {\paren {\ds \sum_{n \mathop = 1}^\infty \dfrac 1 {n^2} - \sum_{n \mathop = 1}^\infty \dfrac {t^n} {n^2} } } {1 - t} \rd t + \gamma \cdot \paren {1 - \map \zeta 2} - \map \zeta 3\) | Tonelli's Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_{\to 0}^{\to 1} \dfrac {\map \zeta 2 - \map {\Li_2} t } {1 - t} \rd t + \gamma \cdot \paren {1 - \map \zeta 2} - \map \zeta 3\) | Definition of Riemann Zeta Function and Power Series Expansion for Spence's Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \map \zeta 3 + \gamma \cdot \paren {1 - \map \zeta 2} - \map \zeta 3\) | Definite Integral from $0$ to $1$ of $\dfrac {\map \zeta 2 - \map {\Li_2} x} {1 - x}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \gamma \cdot \paren {1 - \map \zeta 2} + \map \zeta 3\) |
$\blacksquare$