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Ramanujan's lost notebook: part I

By: Andrews, George E.
Contributor(s): Berndt, Bruce C [Co author].
Material type: materialTypeLabelBookPublisher: New York Springer 2005Description: xiv, 437p. With index.ISBN: 9781493976256.Subject(s): Ramanujan Aiyangar, Srinivasa, 1887-1920 | Mathematics | q-continued FractionsDDC classification: 510 Summary: This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17. Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook. https://www.springer.com/in/book/9780387255293
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Table of contents
Inroduction

Rogers-Ramanujan Continued Fraction and Its Modular Properties

Explicit Evaluations of the Rogers-Ramanujan Continued Fraction

A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions

The Rogers-Ramanujan Continued Fraction and Its Connections with Partitions and Lambert Series

Finite Rogers-Ramanujan Continued Fractions

Other q-continued Fractions

Asymptotic Formulas for Continued Fractions

Ramanujan’s Continued Fraction for (q2;q3)∞/(q;q3)∞

The Rogers-Fine Identity

An Empirical Study of the Rogers-Ramanujan Identities

Rogers-Ramanujan-Slater Type Identities

Partial Fractions

Hadamard Products for Two q-Series

Integrals of Theta Functions

Incomplete Elliptic Integrals

Infinite Integrals of q-Products

Modular Equations in Ramanujan’s Lost Notebook

Fragments on Lambert Series

This volume is the first of approximately four volumes devoted to providing statements, proofs, and discussions of all the claims made by Srinivasa Ramanujan in his lost notebook and all his other manuscripts and letters published with the lost notebook. In addition to the lost notebook, this publication contains copies of unpublished manuscripts in the Oxford library, in particular, his famous unpublished manuscript on the partition and tau-functions; fragments of both published and unpublished papers; miscellaneous sheets; and Ramanujan's letters to G. H. Hardy, written from nursing homes during Ramanujan's final two years in England. This volume contains accounts of 442 entries (counting multiplicities) made by Ramanujan in the aforementioned publication. The present authors have organized these claims into eighteen chapters, containing anywhere from two entries in Chapter 13 to sixty-one entries in Chapter 17. Most of the results contained in Ramanujan's Lost Notebook fall under the purview of q-series. These include mock theta functions, theta functions, partial theta function expansions, false theta functions, identities connected with the Rogers-Fine identity, several results in the theory of partitions, Eisenstein series, modular equations, the Rogers-Ramanujan continued fraction, other q-continued fractions, asymptotic expansions of q-series and q-continued fractions, integrals of theta functions, integrals of q-products, and incomplete elliptic integrals. Other continued fractions, other integrals, infinite series identities, Dirichlet series, approximations, arithmetic functions, numerical calculations, diophantine equations, and elementary mathematics are some of the further topics examined by Ramanujan in his lost notebook.

https://www.springer.com/in/book/9780387255293

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