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An invitation to q-series : from Jacobi's triple product identity to Ramanujan's "most beautiful identity" /
The aim of these lecture notes is to provide a self-contained exposition of several fascinating formulas discovered by Srinivasa Ramanujan. Two central results in these notes are: (1) the evaluation of the Rogers-Ramanujan continued fraction -- a result that convinced G H Hardy that Ramanujan was a...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Singapore :
World Scientific Pub Co.,
©2011.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Introduction
- Part I: Jacobi's triple product identity ; First proof (via functional equation)
- Second proof (via Gaussian polynomials and the q-binomial theorem)
- Some applications
- The Boson-Fermion correspondence
- Macdonald's identities
- Part II: The Rogers-Ramanujan identitites ; First proof (via functional equation)
- Second proof (involving Gaussian polynomials and difference equations)
- Third proof (via Bailey's lemma)
- Excursus : mock theta functions
- Part III: The Rogers-Ramanujan continued fraction ; A list of theorems to be proven
- The evaluation of the Rogers-Ramanujan continued fraction
- A "difficult and deep" identity
- A remarkable identity from the Lost Notebook and cranks
- A differential equation for the Rogers-Ramanujan continued fraction
- Part IV: From the "most beautiful identity" to Ramanujan's congruences ; Proofs of the "most beautiful identity"
- Ramanujan's congruences I : analytical methods
- Ramanujan's congruences II : an introduction to t -cores
- Ramanujan's congruences III : more congruences
- Excursus : modular forms and more congruences for the partition function.