Power Series Solutions of Laguerre Differential Equation

[vjraghavan]

I was going through http://mathworld.wolfram.com/LaguerreDifferentialEquation.html" 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?

 

[HallsofIvy]

Yes, the power series converges, between -1 and 1. A solution at a regular singular point may require the Frobenius method or it may not. That is, it may have a regular power series or it may not. Yes, there is a second independent solution. It will, if I remember correctly, involve a power series time log(x). In any case, it is not defined at 0 and, since the Laguerre equation typically is derived from a problem on the interval [-1, 1] (often from a circularly symmetric situation), that solution tends to be ignored.