Conjectured generating-function structure of the coefficients $\tilde{c}_{2k-1,m}$

Let $a$, $C$, and $\tilde{c}_{2k-1,m}$ be the parameters and coefficients in the asymptotic expansion, and let $P_{k,m}^l(t)$ be polynomials in $t$. For $l\geq0$, define the falling factorial by

$$(m-1)_l=(m-1)(m-2)\cdots(m-l),$$

with $(m-1)_0=1$.

Generating-structure conjecture. For $m\in\mathbb{N}$,

$$\tilde{c}_{2k-1,m}=(-1)^{m-k-1}C^{m-k-2}\sum_{l=0}^{k+1}P_{k,m}^l(a^2)(m-1)_lC^{k+1-l},$$

where each $P_{k,m}^l(t)$ is a polynomial of degree $\lfloor\frac{k}{2}\rfloor$ and has positive rational coefficients.

The formula is motivated by the pole structure of generating functions inferred from their recurrence relations. It has been checked against computed coefficients, but the supplied text does not establish it in general.