The Wolfram functions site includes (without reference) the identity (source)
$$L_0(z)=\frac{4}{\pi}\sum_{k=0}^\infty \frac{I_{2k+1}(z)}{2k+1}$$
where $L_0(z)$ is the modified Struve function of order zero and $I_n(z)$ is the $n$th modified Bessel function of the second kind. But from my work in another question, I think this is incorrect as written and should include $(-1)^k$, i.e., it should be alternating. Can someone provide a reference to either result or a direct proof of such?
Best Answer
By http://dlmf.nist.gov/11.2.E2, http://dlmf.nist.gov/11.4.E21 and http://dlmf.nist.gov/10.27.E6, we have $$ \mathbf{L}_0 (z) = - \mathrm{i}\mathbf{H}_0 (\mathrm{i}z) = - \mathrm{i}\frac{4}{\pi }\sum\limits_{k = 0}^\infty {\frac{{J_{2k + 1} (\mathrm{i}z)}}{{2k + 1}}} = \frac{4}{\pi }\sum\limits_{k = 0}^\infty {( - 1)^k \frac{{I_{2k + 1} (z)}}{{2k + 1}}} . $$