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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# Power Series Solutions of Laguerre Differential Equation

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