I was going through http://mathworld.wolfram.com/LaguerreDifferentialEquation.html" [Broken] in Wolfram which gives brief details about finding a power series solution of the Laguerre Differential Equation. I was reading the special case when v = 0. I read earlier from Differential Equations by Lomen and Mark that a power series (about x=x0) solution of an ODE exists when all polynomial coefficients are analytic at x=x0. The Laguerre equation has coefficients that are not analytic at x=x0=0 and yet this tries to find series solution around x0 = 0. My questions: 1 Will this power series converge? 2 Should not we be using the Frobenius method to solve this equation? 3 Should not this have two linearly independent solutions?