Conjectured polynomial properties of the coefficients $\tilde{c}_{2k-1,m}$

Let $a$, $b$, and $\tilde{c}_{-1,3}$ be the parameters in the expansion, and write

$$\tilde{c}_{2k-1,m}=b^kP_{k,m}(a,\tilde{c}_{-1,3}),$$

where $P_{k,m}$ is a polynomial with rational coefficients. For $k,m,n\in\mathbb{N}$, the following properties are conjectured.

Polynomial-properties conjecture. First, $a=0$ is a first-order zero of $P_{2n-1,m}(a,\tilde{c}_{-1,3})$, while $P_{2n,m}(0,\tilde{c}_{-1,3})\neq0$. Second, when $m\geqslant k+3$, $\tilde{c}_{-1,3}=0$ is a zero of order $m-k-2$ of $P_{k,m}(a,\tilde{c}_{-1,3})$. Finally,

$$\deg P_{k,m}=k+m-1,\qquad \deg_aP_{k,m}=k,\qquad \deg_{\tilde{c}_{-1,3}}P_{k,m}=m-1.$$

These assertions summarize patterns observed in explicitly computed coefficients. Their validity for all indicated indices is not established in the supplied text.