Details
| Original language | English |
|---|---|
| Pages (from-to) | 3-69 |
| Number of pages | 67 |
| Journal | Jahresbericht der Deutschen Mathematiker-Vereinigung |
| Volume | 125 |
| Issue number | 1 |
| Early online date | 18 Aug 2022 |
| Publication status | Published - Mar 2023 |
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.
Keywords
- Formal power series, Jacobi’s triple product, MacMahon’s master theorem, Partitions, Ramanujan, Stirling numbers
ASJC Scopus subject areas
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In: Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 125, No. 1, 03.2023, p. 3-69.
Research output: Contribution to journal › Review article › Research › peer review
}
TY - JOUR
T1 - An Invitation to Formal Power Series
AU - Sambale, Benjamin
N1 - Funding Information: I thank Diego García Lucas, Till Müller and Alexander Zimmermann for proofreading. The work is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/3-1).
PY - 2023/3
Y1 - 2023/3
N2 - 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.
AB - 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.
KW - Formal power series
KW - Jacobi’s triple product
KW - MacMahon’s master theorem
KW - Partitions
KW - Ramanujan
KW - Stirling numbers
UR - http://www.scopus.com/inward/record.url?scp=85136261573&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2205.00879
DO - 10.48550/arXiv.2205.00879
M3 - Review article
AN - SCOPUS:85136261573
VL - 125
SP - 3
EP - 69
JO - Jahresbericht der Deutschen Mathematiker-Vereinigung
JF - Jahresbericht der Deutschen Mathematiker-Vereinigung
SN - 0012-0456
IS - 1
ER -