= Solution
Write $\lambda=N$ with $N\in\mathbb Z_{\geq0}$. Then
$$
f(t)=\frac{(t-1)^N}{t^{N+1}}.
$$
Take $\gamma_1$ 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:
$$
y_{\gamma_1}(z)
=2\pi i\,[t^N]\left(e^{zt}(t-1)^N\right).
$$
Only powers $z^0,\ldots,z^N$ 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> $L_N(z)$:
$$
\boxed{y_1(z)\propto L_N(z)}.
$$
This also follows from the recurrence in part (i), because $a_{N+1}=0$ and every later coefficient vanishes.
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