Suppose that $w=\sqrt{R^2 + s^2 -2Rs\cos\theta}$ with $R\ge s>0$, that $p$ is a positive integer and that $j$ is an integer with $0\le j\le p$. Let $I$ and $K$ denote the modified Bessel functions of the first and second kind respectively. I want to prove the following identity (actually I would be happy with the case $j=0$).
\begin{align\*}
\sqrt{\frac{2}{\pi}}&\int\_0^\pi \frac{K\_{j-1/2}(w)}{w^{j-1/2}}\sin^{2p-1}(\theta)\, d\theta \\
&= (-1)^{p-j}(p-1)! 2^p %\\&\qquad
\sum\_{m=0}^{p-j}(-1)^m \binom{p-j}{m} 
 R^{2m} \frac{K\_{p+m-1/2} (R)}{R^{p+m-1/2}}\cdot
 \frac{I\_{m+j-1/2} (s)} { s^{m+j-1/2}}
\end{align\*}
This is very reminiscent of expressions you see in the chapters on addition formulas and definite integrals in Waton's book on Bessel functions, but I have no idea about how to prove it; I have, however, verified it numerically with Sage for small values of $p$ and $j$.


The case that $j=0$ reduces to 
\begin{align\*}\int\_0^\pi &e^{-w}\sin^{2p-1}(\theta)\, d\theta 
\\&= (-1)^{p}(p-1)! 2^p 
\sum\_{m=0}^{p}(-1)^m \binom{p}{m} 
 R^{2m} \frac{K\_{p+m-1/2} (R)}{R^{p+m-1/2}}\cdot
 \frac{I\_{m-1/2} (s)} { s^{m-1/2}},
\end{align\*}
and this is equivalent to the identity that I was asking about in [Proof of an identity involving $\exp(−|x−s|)dx$
over an even sphere](https://mathoverflow.net/questions/249456/proof-of-an-identity-involving-int-exp-x-sdx-over-an-even-sphere).
I'm hoping that my generalization to the above expression, which looks like a typical Bessel function identity, will be more familiar to people and be open to more standard methods of proof.



A Bessel function integral identity involving $\int_0^\pi \frac{K_{j-1/2}(w)}{w^{j-1/2}}\sin^{2p-1}(\theta)\, d\theta$
I have proved this Bessel identity in my paper [The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials](https://arxiv.org/abs/1708.03227). It is Theorem 4 which is proved in Section 3.


