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An invitation to formal power series

Benjamin Sambale

Abstract

This is a lecture 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.

An invitation to formal power series

Abstract

This is a lecture 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.
Paper Structure (9 sections, 66 theorems, 267 equations)

This paper contains 9 sections, 66 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Definitions and basic properties
  3. The toolkit
  4. Laurent series
  5. The main theorems
  6. Applications to combinatorics
  7. Stirling numbers
  8. Multivariate power series
  9. MacMahon's master theorem

Key Result

Lemma 2.2

With the above defined addition and multiplication $(K[[X]],+,\cdot)$ is an integral domain with identity $1$, i. e. $K[[X]]$ is a commutative ring such that $\alpha\cdot\beta\ne 0$ for all $\alpha,\beta\in K[[X]]\setminus\{0\}$. Moreover, $K$ and $K[X]$ are subrings of $K[[X]]$.

Theorems & Definitions (187)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Example 2.6
  • Definition 2.7
  • Lemma 2.8
  • ...and 177 more