Combinatorial Sums and Identities Involving Generalized Sum-of-Divisors Functions with Bounded Divisors

2017/04/19 by Maxie D. Schmidt, Schmidt, Maxie D.
Engineering · Physics and Astronomy · #(2010): 30B50 #11B37 #11B83 #11K65 #11N64 #Experimental and Theoretical Physics Studies #FOS: Mathematics #Number Theory (math.NT) #Sports Dynamics and Biomechanics

paper · pdf · doi:10.48550/arxiv.1704.05595

openalex publication_date 2017/04/19 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

The class of Lambert series generating functions (LGFs) denoted by L_α(q) formally enumerate the generalized sum-of-divisors functions, σ_α(n) = ∑_d|n d^α, for all integers n ≥ 1 and fixed real-valued parameters α≥ 0. We prove new formulas expanding the higher-order derivatives of these LGFs. The results we obtain are combined to express new identities expanding the generalized sum-of-divisors functions. These new identities are expanded in the form of sums of polynomially scaled multiples of a related class of divisor sums depending on n and α.

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