| Contents |
The Rogers-Ramanujan Continued Fraction and Its Modular Properties -- Two-Variable Generalizations of (1.1.10) and (1.1.11) -- Hybrids of (1.1.10) and (1.1.11) -- Factorizations of (1.1.10) and (1.1.11) -- Modular Equations -- Theta-Function Identities of Degree 5 -- Refinements of the Previous Identities -- Identities Involving the Parameter k = R(q)R[superscript 2](q[superscript 2]) -- Other Representations of Theta Functions Involving R(q) -- Explicit Formulas Arising from (1.1.11) -- Explicit Evaluations of the Rogers-Ramanujan Continued Fraction -- Explicit Evaluations Using Eta-Function Identities -- General Formulas for Evaluating R [characters not reproducible] and S [characters not reproducible] -- Page 210 of Ramanujan's Lost Notebook -- Some Theta-Function Identities -- Ramanujan's General Explicit Formulas for the Rogers-Ramanujan Continued Fraction -- A Fragment on the Rogers-Ramanujan and Cubic Continued Fractions -- The Rogers-Ramanujan Continued Fraction -- The Theory of Ramanujan's Cubic Continued Fraction -- Explicit Evaluations of G(q) -- The Rogers-Ramanujan Continued Fraction and Its Partitions and Lambert Series -- Connections with Partitions -- Further Identities Involving the Power Series Coefficients of C(q) and 1/C(q) -- Generalized Lambert Series -- Further q-Series Representations for C(q) -- Finite Rogers-Ramanujan Continued Fractions -- Finite Rogers-Ramanujan Continued Fractions -- A generalization of Entry 5.2.1 -- Class Invariants -- A Finite Generalized Rogers-Ramanujan Continued Fraction. |
| Contents |
Other q-continued Fractions -- The Main Theorem -- A Second General Continued Fraction -- A Third General Continued Fraction -- A Transformation Formula -- Zeros -- Two Entries on Page 200 of Ramanujan's Lost Notebook -- An Elementary Continued Fraction -- Asymptotic Formulas for Continued Fractions -- The Main Theorem -- Two Asymptotic Formulas Found on Page 45 of Ramanujan's Lost Notebook -- An Asymptotic Formula for R(a,q) -- Ramanujan's Continued Fraction for (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity] -- A Proof of Ramanujan's Formula (8.1.2) -- The Special Case a = w of (8.1.2) -- Two Continued Fractions Related to (q[superscript 2];q[superscript 3])[infinity]/(q;q[superscript 3])[infinity] -- An Asymptotic Expansion -- The Rogers-Fine Identity -- Series Transformations -- The Series [characters not reproducible] -- The Series [characters not reproducible] -- The Series [characters not reproducible] -- An Empirical Study of the Rogers-Ramanujan Identities -- The First Argument -- The Second Argument -- The Third Argument -- The Fourth Argument -- Rogers-Ramanujan-Slater-Type Identities -- Identities Associated with Modulus 5 -- Identities Associated with the Moduli 3, 6, and 12 -- Identities Associated with the Modulus 7 -- False Theta Functions -- Partial Fractions -- The Basic Partial Fractions -- Applications of the Partial Fraction Decompositions -- Partial Fractions Plus -- Related Identities -- Remarks on the Partial Fraction Method -- Hadamard Products for Two q-Series. |
| Contents |
Stieltjes-Wigert Polynomials -- The Hadamard Factorization -- Some Theta Series -- A Formal Power Series -- The Zeros of K[subscript infinity](zx) -- Small Zeros of K[subscript infinity](z) -- A New Polynomial Sequence -- The Zeros of p[subscript n](a) -- A Theta Function Expansion -- Ramanujan's Product for p[subscript infinity](a) -- Integrals of Theta Functions -- Preliminary Results -- The Identities on Page 207 -- Integral Representations of the Rogers-Ramanujan Continued Fraction -- Incomplete Elliptic Integrals -- Preliminary Results -- Two Simpler Integrals -- Elliptic Integrals of Order 5 (I) -- Elliptic Integrals of Order 5 (II) -- Elliptic Integrals of Order 5 (III) -- Elliptic Integrals of Order 15 -- Elliptic Integrals of Order 14 -- An Elliptic Integral of Order 35 -- Constructions of New Incomplete Elliptic Integral Identities -- Infinite Integrals of q-Products -- Proofs -- Modular Equations in Ramanujan's Lost Notebook -- Eta-Function Identities -- Summary of Modular Equations of Six Kinds -- A Fragment on Page 349 -- Fragments on Lambert Series -- Entries from the Two Fragments -- Location Guide -- Provenance. |
| Abstract |
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. |
| Bibliography note | Includes bibliographical references and indexes. |
| Genre/form | Biography. |
| LCCN | 2005923547 |
| ISBN | 9780387255293 (pt. 1 ; hardcover) |
| ISBN | 038725529X (pt. 1 ; hardcover) |
| ISBN | 9780387777658 (pt. 2 ; hardcover) |
| ISBN | 0387777652 (pt. 2 ; hardcover) |
| ISBN | 9781461438090 (pt. 3 ; hardcover) |
| ISBN | 1461438098 (pt. 3 ; hardcover) |
| ISBN | 9781461440802 (pt. 4 ; hardcover) |
| ISBN | 1461440807 (pt. 4 ; hardcover) |
| ISBN | 9783319778327 (pt. 5 ; hardcover) |
| ISBN | 3319778323 (pt. 5 ; hardcover) |
| ISBN | (pt. 1 ; ebook) |
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| ISBN | (pt. 2 ; ebook) |
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| ISBN | (pt. 3 ; ebook) |
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| ISBN | (pt. 4 ; ebook) |
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| ISBN | (pt. 5 ; ebook) |
| ISBN | (pt. 5 ; ebook) |
| Standard identifier# |
9780387255293 |
