Dirichlet Beta Function at Odd Positive Integers

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Theorem

$\map \beta {2 n + 1} = \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}$

where:

$\beta$ denotes the Dirichlet beta function
$E_n$ denotes the $n$th Euler number
$n$ is a non-negative integer.


Proof

\(\displaystyle \map \beta {2 n + 1}\) \(=\) \(\displaystyle \dfrac 1 {4^{2 n + 1} } \paren {\map \zeta {2 n + 1, \frac 1 4} - \map \zeta {2 n + 1, \frac 3 4} }\) Dirichlet Beta Function in terms of Hurwitz Zeta Function
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 {4^{2 n + 1} } \paren { \dfrac {\map {\psi_{2 n} } {\dfrac 1 4} - \map {\psi_{2 n} } {\dfrac 3 4} } { \paren {-1}^{2 n + 1} \map \Gamma {2 n + 1} } }\) Polygamma Function in terms of Hurwitz Zeta Function
\(\displaystyle \) \(=\) \(\displaystyle -\dfrac 1 {4^{2 n + 1} \paren {2 n}!} \paren {\map {\psi_{2 n} } {\dfrac 1 4} - \map {\psi_{2 n} } {\dfrac 3 4} }\)
\(\displaystyle \) \(=\) \(\displaystyle - \dfrac 1 {4^{2 n + 1} \paren {2 n}!} \paren {-\pi \valueat {\frac {\d^{2 n} } {\d z^{2 n} } \cot \pi z} {z \mathop = \frac 1 4} }\) Polygamma Reflection Formula for $z = \dfrac 1 4$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac \pi {4^{2 n + 1} \paren {2 n}!} \valueat {\dfrac {\d^{2 n} } {\d z^{2 n} } \cot \pi z} {z \mathop = \frac 1 4}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac \pi {4^{2 n + 1} \paren {2 n}!} \paren {-1}^n \paren {2 \pi}^{2 n} E_{2 n}\) $\cot \pi z$ $2n$th derivative at $\dfrac 1 4$
\(\displaystyle \) \(=\) \(\displaystyle \paren {-1}^n \dfrac { E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}\)

$\blacksquare$