Gazeau et al.'s recurrence conjecture for exponential-polynomial coefficients

Let $p\geq 2$, let $a_i>0$ for $2\leq i\leq p$, and define

$$x_n!=\prod_{k=1}^{n}x_k.$$

Suppose that the numbers $x_i$ are determined by

$$\exp\left(t+\sum_{i=2}^{p}\frac{a_i}{i}t^i\right)=\sum_{n\geq 0}\frac{t^n}{x_n!}.$$

Gazeau et al.'s conjecture. These numbers satisfy the recurrence relation

$$x_n=\frac{n+1}{1+\sum_{i=2}^{p}a_i\frac{x_n!}{x_{n-i+1}!}}.$$

This is the strong conjecture concerning the coefficients of the Taylor expansion of the exponential of a polynomial; the paper proves it using properties of the Gould–Hopper polynomials, and thereby establishes the corresponding weak conjecture as a special case.