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Factorization theorems for generalized Lambert series and applications

  • Mircea Merca
  • Maxie D. SchmidtEmail author
Article

Abstract

We prove new variants of the Lambert series factorization theorems studied by the authors which correspond to a more general class of Lambert series expansions of the form \(L_a(\alpha , \beta ; q) := \sum _{n \ge 1} a_n q^{\alpha n-\beta } / (1-q^{\alpha n-\beta })\) for integers \(\alpha , \beta \) defined such that \(\alpha \ge 1\) and \(0 \le \beta < \alpha \). Applications of the new results in the article are given to restricted divisor sums over several classical special arithmetic functions which define the cases of well-known, so-termed “ordinary” Lambert series expansions cited in the introduction. We prove several new forms of factorization theorems for Lambert series over a convolution of two arithmetic functions which similarly lead to new applications relating convolutions of special multiplicative functions to partition functions and n-fold convolutions of one of the special functions.

Keywords

Lambert series Factorization theorem Matrix factorization Partition function Multiplicative function 

Mathematics Subject Classification

11A25 11P81 05A17 05A19 

Notes

Acknowledgements

The authors thank the referees for their helpful insights and comments on preparing the manuscript.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Academy of Romanian ScientistsBucharestRomania
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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