On Hypergeometric Solutions of Second Order Linear Differential Equations

Erdal Imamoglu (FSU)

# Abstract

Given a second order linear differential operator $$L$$ with rational function coefficients, we want to find a hypergeometric solution of the form $\exp (\int r\, dx)\; {}_2 F_1(a,b,c;f).$ This form is both more and less general than in prior work. In prior work, solutions involving a sum of two $${}_2F_1$$’s were also considered, however, $$f$$ was restricted to rational functions, while our method allows algebraic functions.

The crucial part is to find $$f$$. By comparing the quotients of formal solutions, we can compute $$f$$ if we know the value of a certain constant $$c$$. We have no direct formula for $$c$$; to obtain it with a finite computation, we take a prime number $$p$$. Then, for each $$c=1,\dots,p-1$$ we try to compute $$f$$ modulo $$p$$. If this succeeds, then we lift $$f$$ modulo a power of $$p$$, and try reconstruction.