= Solution
The <Fresnel convolution of free-particle time slices> follows by completing the square:
$$
\frac{(x_k-x_0)^2}{k}+(x_{k+1}-x_k)^2
=\frac{k+1}{k}\left(x_k-\frac{x_0+kx_{k+1}}{k+1}\right)^2
+\frac{(x_{k+1}-x_0)^2}{k+1}.
$$
The second term is independent of $x_k$. The shifted <Fresnel integral> in the first term gives
$$
\boxed{\sqrt{\frac{2\pi i\hbar\epsilon}{m}\frac{k}{k+1}}\,
\exp\left[\frac{im(x_{k+1}-x_0)^2}
{2\hbar\epsilon(k+1)}\right].}
$$
Here $k$ is the positive slice index, not a <wave number>. The phase is fixed by the same convergence prescription as the short-time kernel.
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