The Frobenius problem for generalized repunit numerical semigroups

In this paper, we introduce and study the numerical semigroups generated by $\{a_1, a_2, \ldots \} \subset \mathbb{N}$ such that $a_1$ is the repunit number in base $b > 1$ of length $n > 1$ and $a_i - a_{i-1} = a\, b^{i-2},$ for every $i \geq 2$, where $a$ is a positive integer relatively prime with $a_1$. These numerical semigroups generalize the repunit numerical semigroups among many others. We show that they have interesting properties such as being homogeneous and Wilf. Moreover, we solve the Frobenius problem for this family, by giving a closed formula for the Frobenius number in terms of $a, b$ and $n$, and compute other usual invariants such as the Ap\'ery sets, the genus or the type.