Congruences for sums involving rkk

We primarily investigate congruences modulo p for finite sums of the form ∑ₖrkkxᵏ/k over the ranges $0<k<p$ and $0<k<p/r$, where p is a prime larger than the positive integer r. Here x is an indeterminate, thus allowing specialization to numerical congruences where x takes certain algebraic numbers as values. We employ two different approaches that have complementary strengths. In particular, we obtain congruences modulo p² for the sum ∑0<k<prkkxᵏ, expressed in terms of finite polylogarithms of certain quantities related to x.