[Paper Review] Combinatorial Sums and Identities Involving Generalized Divisor Functions with Bounded Divisors

Maxie D. Schmidt|arXiv (Cornell University)|Apr 19, 2017

Advanced Mathematical Identities6 references3 citations

TL;DR

This paper derives new identities for generalized divisor functions σα(n) by analyzing higher-order derivatives of Lambert series generating functions Lα(q). The key contribution is a polynomially scaled expansion of σα(n) in terms of bounded divisor sums, offering a novel analytical framework for studying arithmetic functions with applications in number theory and modular forms.

ABSTRACT

The class of Lambert series generating functions (LGFs) denoted by $L_{\alpha}(q)$ formally enumerate the generalized sum-of-divisors functions, $\sigma_{\alpha}(n) = \sum_{d|n} d^{\alpha}$, for all integers $n \geq 1$ and fixed real-valued parameters $\alpha \geq 0$. 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 $\alpha$.

Motivation & Objective

• To derive explicit formulas for higher-order derivatives of Lambert series generating functions Lα(q) that enumerate σα(n).
• To establish new combinatorial identities expressing σα(n) as sums of polynomially scaled divisor sums.
• To analyze the structure of generalized divisor functions σα(n) under bounded divisor constraints.
• To provide a systematic method for expanding σα(n) using generating function derivatives and divisor sum decompositions.

Proposed method

• The paper employs formal differentiation of Lambert series generating functions Lα(q) = ∑_{n≥1} σα(n) q^n to derive higher-order derivative expansions.
• It introduces a transformation that expresses the derivatives of Lα(q) as combinations of divisor sums with polynomial coefficients depending on α and n.
• The method relies on manipulating the Dirichlet series structure of σα(n) through generating function techniques.
• Polynomial scaling factors are derived from the Taylor expansion of the generating function's derivatives around q = 0.
• The resulting identities are constructed by matching coefficients in the power series expansion of the differentiated Lα(q).
• The framework is generalized to arbitrary real α ≥ 0, enabling broad applicability to arithmetic functions.

Experimental results

Research questions

• RQ1 How can higher-order derivatives of Lambert series generating functions Lα(q) be systematically expanded to reveal new identities for σα(n)?
• RQ2 What is the structure of the polynomially scaled divisor sum expansions that represent σα(n) for fixed α and n?
• RQ3 How do bounded divisor constraints influence the form and convergence of these generalized divisor function identities?
• RQ4 Can the derivative-based approach yield closed-form expressions for σα(n) in terms of elementary divisor sums and polynomials?
• RQ5 What is the role of the parameter α in shaping the combinatorial structure of the derived identities?

Key findings

• The paper derives a closed-form expression for the k-th derivative of Lα(q), which is expressed as a sum over divisor functions with polynomial coefficients in k and n.
• New identities are established that represent σα(n) as a linear combination of divisor sums, each scaled by a polynomial in n and α.
• The derived expansions are valid for all real α ≥ 0 and all integers n ≥ 1, ensuring broad applicability.
• The method reveals a recursive structure in the coefficients of the divisor sum expansions, linking higher-order derivatives to lower-order divisor functions.
• The identities demonstrate that σα(n) can be decomposed into a finite sum of weighted divisor sums with rational polynomial weights.
• The framework provides a systematic way to generate identities for σα(n) by choosing different orders of differentiation on Lα(q).