At least one power series solution will be obtained when applying the Frobenius method if the expansion point is an ordinary, or regular, singular point. The number of roots is given by the roots of the indicial equation.
Fuchs's Theorem
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References
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 462-463, 1985.Referenced on Wolfram|Alpha
Fuchs's TheoremCite this as:
Weisstein, Eric W. "Fuchs's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FuchssTheorem.html