 Solution

Write $\lambda =N$ with $N\in {\mathbb{Z}}_{\ge 0}$. ThenTake ${\gamma }_1$ to be a small positively oriented circle around $t=0$. For ${\gamma }_2$, start at $t=1$, travel to $-\infty$, and avoid the pole at zero by a fixed upper or lower detour. The endpoint factor vanishes at $t=1$, and the exponential controls $-\infty$.

The finite contour extracts the residue at zero:Only powers $z^0,\dots ,z^N$ occur, and the coefficient of $z^N$ is nonzero. Thus this solution is a polynomial of degree $N$ and, up to normalization, is the Laguerre polynomial $L_N(z)$:This also follows from the recurrence in part (i), because $a_{N+1}=0$ and every later coefficient vanishes.