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.