An Invitation to Formal Power Series

authored by

Benjamin Sambale

Abstract

This is an account on the theory of formal power series developed entirely without any analytic machinery. Combining ideas from various authors we are able to prove Newton’s binomial theorem, Jacobi’s triple product, the Rogers–Ramanujan identities and many other prominent results. We apply these methods to derive several combinatorial theorems including Ramanujan’s partition congruences, generating functions of Stirling numbers and Jacobi’s four-square theorem. We further discuss formal Laurent series and multivariate power series and end with a proof of MacMahon’s master theorem.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

Type

Review article

Journal

Jahresbericht der Deutschen Mathematiker-Vereinigung

Volume

125

Pages

3-69

No. of pages

67

ISSN

0012-0456

Publication date

03.2023

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Mathematics(all)

Electronic version(s)

https://doi.org/10.48550/arXiv.2205.00879 (Access: Open)
https://doi.org/10.1365/s13291-022-00256-6 (Access: Open)