Let us consider $e^k_n$ defined for $k, n \in \mathbb{N}$ such that
$$ e^0_n = \delta \quad \forall n \in \mathbb{N} $$
and
$$ e_{n+1}^k = \delta \frac{\gamma^k}{(k-1)!} \sum_{i=0}^n \prod_{l=1}^{k-1}(i+l)\beta^i $$
What is the double recurrence relation verified by $e_n^k$ ?
📜 Just use the classical delimiter
$$ ... $$for equations and$ ... $for inline latex.
This is something you can compute using the generating function :
$$\rho_{k}(\zeta) := \sum_{n=0}^{\infty} e_{n+1}^{k} \zeta^{n+1}$$
⚠️ When using exponent and index for one variable, always put the index before the exponent, or else Mathjax won't render the latex formula correctly !
For instance :
e_{n+1}^k -> OK
e^k_{n+1} -> NOT OK