Ratios of modified Bessel functions with different second arguments

Question

With regards to the first modified Bessel function $I_\nu(x)$, much appears to be known about ratios with differing first arguments, i.e. ratios of the form $I_{\nu + 1}(x) / I_\nu(x)$ have certain asymptotic expansions and representations in terms of partial fractions. My question is, is anything at all known about ratios of the form $I_\nu(x) / I_\nu(y)$? My particular interest is in being able to numerically evaluate ratios of these forms efficiently, and I am wondering if any work has been done on this - searching online turned up no results, but I may not have been searching for the right terms. I am only interested in integer $\nu$, if that that is of any help.

Answer 1

The asymptotic expansions of the modified Bessel functions are well known. (Though I have slightly adjusted the formulas from DLMF for enhanced clarity.) For example, $$I_{\nu}(z)\asymp \frac{\exp z}{(2\pi z)^{1/2}} \sum_{m=0}^\infty a_m(\nu)z^{-m}\\ \text{for}~~\operatorname{Re}z>0$$ With $$a_m(\nu)=\frac{1}{2^mm!}~\prod_{j=0}^{m-1}\left(\frac{1}{4}-(\nu+j)^2\right)$$ We have $$a_0(\nu)=1$$ So a crude but nonetheless robust approximation is $$I_\nu(z)\asymp \frac{\exp z}{(2\pi z)^{1/2}}$$ And so, for large $x,y$ (and, crucially, $\nu \ll x,y$) one can roughly approximate $$\frac{I_\nu(x)}{I_\nu(y)}\approx e^{x-y}\sqrt{x/y}$$ This approximation is very crude, but it will typically give you a good "order of magnitude" estimate.

For more precise asymptotics you will need to get more creative. Perhaps useful is the integral representation for $n\in\mathbb Z$: $$I_n(z)=\frac{1}{\pi}\int_0^\pi \exp(z\cos\theta)\cos(n\theta)\mathrm d\theta$$ Though I must admit I was unable to make any progress with this.