For oblate spheroidal angular and radial functions as defined in Flammer's book: Spheroidal Wave Functions. I found an identity by numerical simulations shown that
$$ \sum_{m=0}^\infty \sum_{n=m}^\infty\left[\frac{S_{mn}(-ic,\pm 1)}{\sqrt{N_{mn}(-ic)}}R_{mn}^{(1)}(-ic,i\xi)\right]^2 = \frac12 $$ for arbitrary $c$ and $\xi$.
Can anyone prove this result?
This identity may play an important role in some problems in physics concerning laws of conservation of energy.
Edit #1:
More detail about the Flammer's definition can be found HERE (click Click here for a description of oblfcn at the bottom of the website).
The definition of $S_{mn}(-ic,\eta)$ reads that $$ S_{mn}(-ic,\eta)=\sum_{r=0,1}^{\infty'} d_r^{mn}(-ic)P_{m+r}^m(\eta) $$ where $P_{m+r}^m(\eta)$ is the associated Legendre function. The $S_{mn}(-ic,\eta)$ form an orthogonal set on the interval $(-1,1)$, here the normaliation factor is $$ N_{mn}(-ic)=2\sum_{r=0,1}^{\infty'}\frac{(r+2m)![d_r^{mn}(-ic)]^2}{(2r+2m+1)r!} $$
The definition of $R_{mn}^{(1)}(-ic,i\xi)$ reads that $$ R_{mn}^{(1)}(-ic,i\xi) = \left(\frac{\xi^2+1}{\xi^2}\right)^{m/2} \frac{\sum_{r=0,1}^{\infty'} i^{r+m-n} d_r^{mn}(-ic) j_{m+r}(c\xi) \frac{(r+2m)!}{r!}}{\sum_{r=0,1}^{\infty'} d_r^{mn}(-ic)\frac{(r+2m)!}{r!}} $$ where $j_m(x)$ is the first kind of spherical Bessel function.