Restricted partition functions and the $r$-log-concavity of quasi-polynomial-like functions

Let $\mathcal A=(a_i)_{i=1}^\infty $ be a non-decreasing sequence of positive integers and let $k$ be a fixed positive integer. For an arbitrary integer $n$, the restricted partition function $p_\mathcal A(n,k)$ enumerates all the partitions of $n$ whose parts belong to the multiset $\{a_1,\ldots ,a_k\}$. In this paper we investigate some generalizations of the log-concavity of $p_{\mathcal A}(n,k)$. We deal with both some basic extensions like, for instance, the strong log-concavity and a more intriguing challenge that is the $r$-log-concavity of both quasi-polynomial-like functions in general, and the restricted partition function in particular. For each of the problems, we present an efficient solution.