Is there a closed form for $$f(a)=\sum_{k=1}^\infty\frac{1}{(2^k-1)^a},$$ where $0<a\in\mathbb{R}$.

My attempts so far have considered $a\in\mathbb{N}$, which appears to give finite sums of the q-Polygamma function, e.g.

$$f(4)=\frac{\psi _{\frac{1}{2}}^{(3)}(1)}{6 \log ^42}+\frac{\psi _{\frac{1}{2}}^{(2)}(1)}{\log ^32}+\frac{11 \psi _{\frac{1}{2}}^{(1)}(1)}{6 \log ^22}+\frac{\psi _{\frac{1}{2}}^{(0)}(1)}{\log 2}-1.$$

"Closed form" here could be an integral, or something akin to hypergeometric functions.

Update: The Erdős–Borwein constant is related.


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