Applications of Faà di Bruno's formula to partition traces

2025/07/01 by Toshiki Matsusaka, Matsusaka, Toshiki · 2 citations
Mathematics · #05A15 #05A17 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2507.00404

openalex publication_date 2025/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We revisit several partition-theoretic generating functions, including the theta quotients from Ramanujan's lost notebook, MacMahon's partition functions, and reciprocal sums of parts in partitions, through the lens of the classical Faà di Bruno formula. This approach offers a unified and natural reinterpretation of known results and provides a systematic framework for deriving new identities of a similar type.

Citations

• Quasimodularity and Limiting Behavior for Variations of MacMahon Series
• Modularity of moments of reciprocal sums for partitions into distinct parts
• Some topological genera and Jacobi forms
• Quasi-modularity in MacMahon partition variants and prime detection
• Improved asymptotics for moments of reciprocal sums for partitions into distinct parts
• Further study on MacMahon-type sums of divisors
• Derivatives of theta functions as Traces of Partition Eisenstein series
• MacMahon's sums-of-divisors and their connection to multiple Eisenstein series
• Curious congruences for cyclotomic polynomials II
• De Moivre and Bell polynomials
• Coefficients and higher order derivatives of cyclotomic polynomials: old\n and new
• Quasi-modularity of generalized sum-of-divisors functions
• MacMahon's sum-of-divisors functions, Chebyshev polynomials, and\n Quasi-modular forms

Cited by

• Multiple \wp-Functions and Their Applications
• Quasimodular forms arising from Jacobi's theta function and special symmetric polynomials

Related

• Andrews Style Partition Identities
• Some partition and analytical identities arising from Alladi, Andrews, Gordon bijections
• Why do partitions occur in Faa di Bruno's chain rule for higher derivatives?
• Partition implications of a new three parameter q-series identity
• Partition analysis and the little Göllnitz identites