= Frobenius series for the Laguerre differential equation
{c}
{title2=$a_{k+1}=\frac{k-\nu}{(k+1)^2}a_k$}
A <Frobenius solution> $y=\sum_{k\geq0}a_kx^{k+\sigma}$ has the repeated <indicial root> $\sigma=0$. Equating coefficients gives the displayed <recurrence relation>. The resulting analytic solution is an <entire function>: its consecutive term ratio has magnitude asymptotic to $|x|/k$. For $\nu=n\geq0$ an integer, the recurrence terminates and $L_n(x)=\sum_{k=0}^n(-1)^k\binom nkx^k/k!$. The second local solution is independent and has a logarithmic term at zero; repeated <indicial roots> do not supply two independent analytic series.
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