Dirichlet beta in terms of Lerch transcendent

Theorem

The following formula holds: $$\beta(x) = 2^{-x} \Phi \left(-1,x,\dfrac{1}{2} \right),$$ where $\beta$ denotes Dirichlet beta and $\Phi$ denotes the Lerch transcendent.

Proof

Starting from the Hadamard Fractional Integral representations of the Lerch Transcendent and the Dirichlet Beta functions, namely,

$$\Phi (z,\alpha ,y) = \sum_{k=0}^\infty\tfrac{z^k}{(y+k)^\alpha},$$

and

$$\beta (\alpha ) = \sum_{k=0}^\infty \tfrac{(-1)^k}{(2k+1)^\alpha}$$

$$2^{-\alpha } \Phi \left(-1,\alpha ,\dfrac{1}{2} \right) = 2^{-\alpha } \sum_{k=0}^\infty\tfrac{(-1)^k}{\left( \dfrac{1}{2}+k\right) ^\alpha} = \beta(\alpha )$$,

and the proof is demonstrated.