Classicality criterion for the odd-indexed Painlevé V solution

Let $l$ be an odd positive integer, and let $a_l^l$ denote the coefficient in the expansion of the solution referred to as Expansion~. Classicality criterion. The solution with odd $l$ is classical if and only if

$$a_l^l=-\frac{1}{l!2^{2l-1}}.$$

The preceding discussion derives this formula for the examples $l=1,3,5$ and indicates that the general assertion follows by mathematical induction using Bäcklund transformations; the source does not provide evidence that the criterion has been proved beyond this argument.