Special function related to a nonlinear ODE

Question

I am interested in finding a special function solution of the following ODE

$ (r S)^{\prime \prime} = - 2 r S^2 $ with initial conditions $ S(0)=1, S^\prime(0) = 0 $.

The method of Frobenius gives the infinite series

$ S= 1 -\frac{{{r}^{2}}}{3}+\frac{{{r}^{4}}}{15}-\frac{11 {{r}^{6}}}{945}+\frac{16 {{r}^{8}}}{8505}-\frac{97 {{r}^{10}}}{334125} +\frac{914 {{r}^{12}}}{21049875} +H.O.T $

and the following recursion among the coefficients:

$ a_n= - \frac{2}{n (n+1)}\sum\limits_{j=0}^{n-2}{\left. a_j a_{n-j-2} \right.}$

The question is: is this function related to the generalized hypergeometric functions?
If yes - what is the relationship?

Answer 1

Let $U=rS$ ,

Then $U''=-\dfrac{2U^2}{r}$ with initial conditions $U(0)=0$ , $U'(0)=1$

Which reduces to an Emden–Fowler equation.