Expansion of a modified Struve function in terms of modified Bessel functions

Question

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?

Answer 1

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 $$ \mathbf{L}_0 (z) = - \mathrm{i}\mathbf{H}_0 (\mathrm{i}z) = - \mathrm{i}\frac{4}{\pi }\sum\limits_{k = 0}^\infty {\frac{{J_{2k + 1} (\mathrm{i}z)}}{{2k + 1}}} = \frac{4}{\pi }\sum\limits_{k = 0}^\infty {( - 1)^k \frac{{I_{2k + 1} (z)}}{{2k + 1}}} . $$

Answer 2

Here is a direct proof. We recall the Jacobi-Anger expansion $$e^{z \sin \theta}=I_0(z)+2\sum_{k=0}^\infty (-1)^k I_{2k+1}(z)\sin((2k+1)\theta)+2\sum_{k=1}^\infty (-1)^k I_{2k}(z)\cos(2k\theta).$$ Evaluating the first $\theta$-derivative at $\theta=0$, we obtain the identity

$$z=\frac{d}{d\theta}\left(e^{z\sin \theta}\right)_{\theta=0} =2\sum_{k=0}^\infty (-1)^k (2k+1)I_{2k+1}(z).$$

We now recall that the modified Bessel function $I_n(z)$ satisfies the differential equation

$$x^2 I_n''(x)+x I_n'(x) -(x^2+n^2)I_n(x)=0.$$ As such, we have

\begin{align} \frac{2}{\pi}z &=\frac{4}{\pi}\sum_{k=0}^\infty (-1)^k (2k+1)I_{2k+1}(z)\\ &=(x^2 D_x^2 +x D_x -x^2)\sum_{k=0}^\infty \frac{4}{\pi}(-1)^k (2k+1)^{-1}I_{2k+1}(z) \end{align} It follows that the sum $$y(z):=\frac{4}{\pi}\sum_{k=0}^\infty (-1)^k (2k+1)^{-1}I_{2k+1}(z)$$ is a solution of the inhomogeneous Bessel equation $$x^2 y''(x)+xy'(x)+x^2 y(x)=\frac{2}{\pi}x.$$ Since all odd-integer modified Bessel functions vanish at zero, it follows that $y$ also vanishes at zero. We now note that the zeroth modified Struve function $L_0(z)$ is defined as the unique solution of the above inhomogeneous Bessel equation which vanishes at $z=0$. Therefore we conclude that indeed

$$L_0(z)=\frac{4}{\pi}\sum_{k=0}^\infty (-1)^k \frac{I_{2k+1}(z)}{2k+1}.$$