$\underline{Intro \;to \;skip}$

In the theory of ellipsoidal harmonics, Lame functions of the second kind $F_n$ arise as the second linearly independent solution (the first being Lame functions of the first kind $E_n$) of Laplace's equation in ellipsoidal coordinates which is the so called Lame's equation. Lame functions of the second kind contain the improper elliptic integral $I_n(\rho)$ and for this reason they are obtainable through numerical approximations e.g. Gauss-Legrendre quadrature. For anyone interested in ellipsoidal harmonics the main bibliographical references (as fas as I know) are Ellipsoidal Harmonics by Dassios, The Theory of Spherical and Ellipsoidal Harmonics by Hobson, An Elementary Treatise on Fourier's Series by Byerly.

$\underline{Question}$

My goal is to find an expansion in powers of 1/ρ (and its first 2 or 3 terms) of the quantity \begin{equation}\label{eq:1} F(\rho,\mu,\nu)=(2n+1)E_n(\rho)E_n(\mu)E_n(\nu)I_n(\rho),\quad \rho \ge h_2\end{equation} where \begin{equation} I_n(\rho)=\int_\rho^{+\infty}\frac{dt}{(E_n(t))^2\sqrt{t^2-h_2^2}\sqrt{t^2-h_3^2}} \end{equation} $\rho,\mu,\nu$ are the ellipsoidal coordinates and \begin{equation} E_n(t)=t^{n-2r}\sum_{i=0}^ra_it^{2i}, \end{equation} where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{odd}. \end{align}

$\underline{What\; have\;I\; tried}$

A first step to solution is to set $t\rightarrow \frac{h_2}{t}$ and this leads to \begin{equation} I_n(\rho)=\int_0^\frac{h_2}{\rho}\frac{t^{2n}/h_2^{2n}}{(E_n(t))^2\sqrt{1-t^2}\sqrt{1-\frac{h_3^2}{h_2^2}t^2}}dt. \end{equation} Using binomial theorem \begin{align} (1-t^2)^{-1/2}&=\sum_{n=0}^{+\infty}(-1)^n\binom{-\frac{1}{2}}{n}t^{2n},\\ (1-\frac{h_3^2}{h_2^2}t^2)^{-1/2}&=\sum_{n=0}^{+\infty}(-1)^n\binom{-\frac{1}{2}}{n}\frac{h_3^{2n}}{h_2^{2n}}t^{2n}, \end{align} and multiplying gives the series expansion \begin{equation}\sum_{n=0}^{+\infty}\sum_{k=0}^n(-1)^n\binom{-\frac{1}{2}}{n-k}\binom{-\frac{1}{2}}{k}\frac{h_3^{2k}}{h_2^{2k+1}}t^{2n}\end{equation} so: \begin{equation}I_n(\rho)=\int_0^{h_2/\rho}\left[\frac{t^{2n}/h_2^{2n}}{(E_n(t))^2 }\sum_{n=0}^{+\infty}\sum_{k=0}^n(-1)^n\binom{-\frac{1}{2}}{n-k}\binom{-\frac{1}{2}}{k}\frac{h_3^{2k}}{h_2^{2k+1}}t^{2n}\right] dt .\end{equation}

$\underline{The\;problem}$

I don t know how to take it from here. The basic problem is that by trying to express everything as a sum in the quantity of F which is the brute force/obvious way, it gets so complicated to perform multiplication between these sums that I don't know if I could find my 2-3 first terms by the resulting series expansion.

Hope you like the subject and offer some help on the matter. Thank you very much :)