Frobenius series for the Laguerre differential equation ($a_{k+1}=\frac{k-\nu }{(k+1{)}^2}{a}_k$)

A Frobenius solution $y={\sum }_{k\ge 0}{a}_k{x}^{k+\sigma }$ 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 $\nu =n\ge 0$ an integer, the recurrence terminates and $L_n(x)={\sum }_{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})x^k/k!$. The second local solution is independent and has a logarithmic term at zero; repeated indicial roots do not supply two independent analytic series.