On certain properties of solutions of second-order linear differential equations

For a second-order linear differential equation with coefficients in a differential fieldL of meromorphic functions, a classical result of C. L. Siegel states that if the logarithmic derivative of every nontrivial solution is transcendental overL, then no nontrivial solution can satisfy a first-order algebraic differential equation overL. In Part 1, we consider a situation for the equation (*)w¨+A(z)w=0 where Siegel's condition is violated, namely where the logarithmic derivative of a nontrivial solution belongs toL. In this case, we obtain a representation for any solution of (*) which satisfies a first-order algebraic differential equation overL, and we illustrate our result with examples for various choices ofA(z) andL. In Part 2, we treat the question of determining when all solutions of (*) satisfy first-order algebraic differential equations overL, and in Part 3, we obtain a refinement of Siegel's classical theorem on Bessel's equation.This research was supported in part by the National Science Foundation (MCS-8002269).