Series of reciprocal powers of figurate numbers

Abstract

We derive closed-form expressions for infinite series of reciprocal powers of figurate numbers, including polygonal numbers, centered polygonal numbers, pyramidal numbers, and centered pyramidal numbers.
Our approach relies on partial fraction decompositions and analytic properties of the Hurwitz zeta and polygamma functions.
In the case of polygonal numbers, we generalize and extend results previously conjectured in the literature. For centered polygonal and pyramidal numbers, we develop systematic methods for evaluating the corresponding reciprocal power sums. Notably, we present what appear to be the first general closed-form formulas for the sums of reciprocal powers of centered pyramidal numbers. Extensive tables of evaluated sums are provided to illustrate the results.