Dirichlet Beta Function at Odd Positive Integers
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Theorem
\(\ds \map \beta {2 n + 1}\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k} {\paren {2 k + 1}^{2 n + 1} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1^{2 n + 1} } - \frac 1 {3^{2 n + 1} } + \frac 1 {5^{2 n + 1} } - \frac 1 {7^{2 n + 1} } + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}\) |
where:
- $\beta$ denotes the Dirichlet beta function
- $E_n$ denotes the $n$th Euler number
- $n$ is a non-negative integer.
Proof
\(\ds \map \beta {2 n + 1}\) | \(=\) | \(\ds \dfrac 1 {4^{2 n + 1} } \paren {\map \zeta {2 n + 1, \frac 1 4} - \map \zeta {2 n + 1, \frac 3 4} }\) | Dirichlet Beta Function in terms of Hurwitz Zeta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 {4^{2 n + 1} } \paren { \dfrac {\map {\psi_{2 n} } {\dfrac 1 4} - \map {\psi_{2 n} } {\dfrac 3 4} } {\paren {-1}^{2 n + 1} \map \Gamma {2 n + 1} } }\) | Polygamma Function in terms of Hurwitz Zeta Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\dfrac 1 {4^{2 n + 1} \paren {2 n}!} \paren {\map {\psi_{2 n} } {\dfrac 1 4} - \map {\psi_{2 n} } {\dfrac 3 4} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds - \dfrac 1 {4^{2 n + 1} \paren {2 n}!} \paren {-\pi \valueat {\frac {\d^{2 n} } {\d z^{2 n} } \cot \pi z} {z \mathop = \frac 1 4} }\) | Polygamma Reflection Formula for $z = \dfrac 1 4$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac \pi {4^{2 n + 1} \paren {2 n}!} \valueat {\dfrac {\d^{2 n} } {\d z^{2 n} } \cot \pi z} {z \mathop = \frac 1 4}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac \pi {4^{2 n + 1} \paren {2 n}!} \paren {-1}^n \paren {2 \pi}^{2 n} E_{2 n}\) | Even Derivatives of Cotangent of Pi Z at One Fourth | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-1}^n \dfrac { E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}\) |
$\blacksquare$
Examples
Dirichlet Beta Function of $1$
- $\map \beta 1 = \dfrac \pi 4 $
Dirichlet Beta Function of $3$
- $\map \beta 3 = \dfrac {\pi^3} {32} $
Dirichlet Beta Function of $5$
- $\map \beta 5 = \dfrac {5 \pi^5} {1536} $