Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. $$\left[\frac{\partial J_\nu(x)}{\partial\nu}\right]_{\nu=1/2}=\sqrt{\frac2{\pi\,x}}\Big(\operatorname{Ci}(2\,x)\sin x -\operatorname{Si}(2\,x)\cos x\Big).$$ In this question I am particularly interested in the case $\nu=0$: $$\begin{array}{} &\left[\frac{\partial J_\nu(x)}{\partial\nu}\right]_{\nu=0}=\frac\pi2Y_0(x), &\left[\frac{\partial Y_\nu(x)}{\partial\nu}\right]_{\nu=0}=-\frac\pi2J_0(x),\\ &\left[\frac{\partial K_\nu(x)}{\partial\nu}\right]_{\nu=0}=0, &\left[\frac{\partial I_\nu(x)}{\partial\nu}\right]_{\nu=0}=-K_0(x).\end{array}$$

Are there similar formulae for the Struve functions $\mathbf H_\nu(x)$, $\mathbf L_\nu(x)$, Anger function ${\bf{J}}_\nu(x)$ and Weber function ${\bf{E}}_\nu(x)$?

Solution 1:

Surprisingly, there are formulae, though a bit complicated (as they involve the Meijer $G$-function). In this paper, Brychkov and Geddes display some formulae for $\mathbf{H}_0^\ast(z)$ and $\mathbf{L}_0^\ast(z)$, where we borrow the "Petiau notation"

$$f_\nu^\ast(z)=\left.\frac{\mathrm d}{\mathrm du}f_u(z)\right|_{u=\nu}$$

from a previous paper by Apelblat.

Here they are:

$$\begin{align} \mathbf{H}_0^\ast(z)&=\frac{\pi}{2}J_0(z)+\frac{2\pi}{z}G_{4,6}^{3,2}\left(\frac{z^2}{4}\mid{{1,1,\frac14,\frac34}\atop{\frac12,1,1,\frac14,\frac12,\frac34}}\right)\\ &=-\frac{\pi}{2}J_0(z)+\frac1{\pi z}G_{2,4}^{3,2}\left(\frac{z^2}{4}\mid{{1,1}\atop{\frac12,1,1,\frac12}}\right) \end{align}$$

and

$$\begin{align} \mathbf{L}_0^\ast(z)&=-K_0(z)-\frac2{z}G_{4,6}^{4,2}\left(\frac{z^2}{4}\mid{{1,1,\frac14,\frac34}\atop{\frac12,\frac12,1,1,\frac14,\frac34}}\right)\\ &=K_0(z)-\frac1{\pi^2 z}G_{2,4}^{4,2}\left(\frac{z^2}{4}\mid{{1,1}\atop{\frac12,\frac12,1,1}}\right) \end{align}$$

These formulae seem to be only valid for $\Re z > 0$, tho. See the linked papers for more general expressions for $\mathbf{H}_\nu^\ast(z)$ and $\mathbf{L}_\nu^\ast(z)$. If there are simpler expressions for the Meijer $G$ terms, I seem to be unable to find them.

As the Anger and Weber functions of integer order are expressible in terms of the usual Bessel and Struve functions, the expressions above and the expressions you are already familiar with can be used to derive expressions for these functions' order derivatives. Nevertheless, see also this other paper by Brychkov and Geddes.