Finite Laguerre contour solution

A finite contour based at an integrable branch point, or a closed contour around an integer-order pole, gives a solution analytic at $z=0$. It is therefore a constant multiple of the unique Frobenius power-series solution there.

For $\lambda =N\ge 0$, the amplitude $(t-1{)}^N/t^{N+1}$ has a pole at zero whose contour residue is a degree-$N$ Laguerre polynomial in $z$. For $\lambda =-m<0$, the amplitude $t^{m-1}/(t-1{)}^m$ has a pole at one whose residue is $e^z$ times a polynomial of degree $m-1$.