Combinatorial and Asymptotic Analysis of Higher-Order Summations

Published March 25, 2026 | Version v1

Thesis Open

This paper presents a bottom-up combinatorial methodology for analyzing higher-order

cumulative summations through the enumeration of nodes in nonlinear tree structures. By

constructing matrix splittings and ascending identities, we present an alternative derivation of

the classical closed-form expressions for higher-order power sums. The termial function,

introduced by Knuth as an analog to factorial, serves as the foundation for this analysis. We

establish a unified framework connecting discrete iterated summations with continuous

extensions, and provide complete algorithmic implementations alongside rigorous mathematical

proofs.

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