Recurrence Relations for k-Fold Nested Power Sums

2025/11/17 by Povolotsky, Alexander R.
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2511.15729

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.

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