The average of the k-fold divisor function \(\begin{aligned} d_k(n) {:}{=} \# \{(a_1, \dots , a_k) \in {\mathbb {Z}}_{\geqslant 0}^k: a_1 \cdots a_k = n\} \end{aligned}\) over arithmetic progressions has been widely studied. The Beatty sequence \(\begin{aligned} {\mathcal {B}}_{\alpha ,\beta } {:}{=} \left( \lfloor \alpha n+\beta \rfloor \right) _{n=1}^\infty \end{aligned}\) is considered as a generalization of arithmetic progressions. In this article, we estimate the average of the k-fold divisor function over Beatty sequences in arithmetic progressions, namely \(\begin{aligned} \sum _{\begin{array}{c} {n \leqslant x, n\in {\mathcal {B}}_{\alpha ,\beta }}\\ {n\equiv a (\textrm{mod}\, r) } \end{array}} d_{k}(n). \end{aligned}\) We obtain the main term and give bounds on the error terms for different k and a refinement for \(k \geqslant 5\) when \(\alpha \) is of finite type. This covers the previous results related to the average of divisor functions over Beatty sequences.
