With great pleasure I read the recent paper of Griffin, Ono, Rolen and Zagier proving the surprising result that the Jensen polynomials $J^{d, n}_\alpha$ for a sequence $\alpha = \{\alpha(0), \alpha(1), \ldots \}$ of real numbers whose growth (?) is controlled in a certain way converges for fixed $d$ to a limiting polynomial of the same degree uniformly on compact subsets of $\mathbb{R}$.

Their main application is to some sequence of real numbers coming from the Riemann Xi function (see also this other MO question) but I already had lots of fun trying to see how this works out for much simpler sequences such as $\alpha(n) = 1$ or $\alpha(n) = 2^n$.

My question is however with the conditions in their main non-RH-related result: theorem 8 which reads:

Suppose that $\{E(n)\}$ and $\{\delta(n)\}$ are positive real sequences with $\delta(n)$ tending to $0$, and that $F(t) = \sum_{i =1}^\infty c_i t^i$ is a formal power series with complex coefficients. For a fixed $d \geq 1$, suppose that there are real sequences $\{C_0(n)\},\ldots,\{C_d(n)\}$, with $\lim_{n \to \infty} C_i(n) = c_i$ for $0 \leq i \leq d$, such that for $0 \leq j \leq d$, we have

$$\frac{\alpha(n+j)}{\alpha(n)} E(n)^{-j} = \sum_{i =0}^d C_i(n) \delta(n)^i j^i + o(\delta(n)^d) \qquad (*)$$ as $n \to \infty$. Then we have:

$$\lim_{n \to \infty} \frac{\delta(n)^{-d}}{\alpha(n)} J^{d, n}_\alpha \left(\frac{\delta(n)X - 1}{E(n)}\right) = H_{F, d}$$

uniformly on compact subsets of $\mathbb{R}$ where $H_{F, d}$ is defined by the generating function $F(−t) e^{Xt}=\sum_{m=0}^\infty H_{F,m}(X) \frac{t^m}{m!}$.

My question is about (*). Hopefully it is clear why I wrote above that I already had fun seeing what this theorem means even for really simple sequences $\alpha$: it is a priori not at all clear what $\delta, E$ or $C_i$ to take and one surprising thing I found is that (unlike their limits $c_i$) the sequences $C_i$ may depend non-trivially on the choice of the fixed value of $d$ even in cases we know a priori that the the limits exist for all $d$.

But managing to find sequence $E, \delta, C_i$ that 'work' is something quite different from understanding what is going on. My question is: what is, intuitively speaking, the set of conditions (*) trying to convey? Is it saying that the sequence $\alpha$ cannot grow too fast? Something else? Is it reasonable to think of finite sum on the right hand side as 'roughly a constant' so that the condition says that $\alpha$ grows more or less as $E^j$ where $E$ is the 'typical' value of $E(n)$. Ugh, as soon as I type it it stops making sense.

Any enlightenment is welcome here.