Recurrence Relations for k-Fold Nested Power Sums

TL;DR

This paper derives explicit recurrence relations for k-fold nested power sums of integers, connecting sums of different powers and nesting levels, based on algebraic identities involving binomial coefficients.

Contribution

It introduces a new recurrence relation for nested power sums, linking lower powers and higher nesting levels, expanding understanding of Faulhaber-type sums.

Findings

Derived explicit recurrence relation for nested sums

Connected new recurrence to existing hypersum relations

Discussed relevance to prior nested sum research

Abstract

We consider the $k$-nested sum of integer powers, $F(n,m,k)$, defined as repeated partial sums of the classical Faulhaber polynomials. We provide an explicit recurrence relation relating $F(n,m,k)$ to sums of lower power $m-1$ and higher nesting level $k+1$. This identity is derived from a core algebraic relation on the binomial coefficients that form the kernel of the nested sum's representation. We discuss the relevance to the 2010 paper by S.~Butler and P.~Karasik, ``A Note on Nested Sums'' (JIS, Vol.~13, Article~10.4.4), which studies nested sums of powers of integers that generalize Faulhaber-type sums. We also discuss the equivalence to a related recurrence previously established in the context of hypersums of powers of integers by J.~L.~Cereceda.