Some New Ramanujan's Modular Equations of Degree 15

Author:

Chuan-Ding Zhang1,Li Yang1

Affiliation:

1. Henan University

Abstract

Abstract Ramanujan in his notebook recorded two modular equations involving multiplier with moduli of degrees (1,7) and (1,23). In this paper, we find some new Ramanujan's modular equations involving multiplier with moduli of degrees (3,5) and (1,15), and give concise proofs by employing Ramanujan's multiplier function equation. Mathematics Subject Classification: 11F20 , 33C05

Publisher

Research Square Platform LLC

Reference7 articles.

1. Berndt, B. C. (1989) Ramanujan{\textquoteright}s Notebooks: Part II. Springer, Germany, 9780387967943, English (US), 10.1007/978-1-4612-4530-8, During the years 1903-1914, Ramanujan recorded many of his mathematical discoveries in notebooks without providing proofs. Although many of his results were already in the literature, more were not. Almost a decade after Ramanujan's death in 1920, G.N. Watson and B.M. Wilson began to edit his notebooks but never completed the task. A photostat edition, with no editing, was published by the Tata Institute of Fundamental Research in Bombay in 1957. This book is the second of four volumes devoted to the editing of Ramanujan's Notebooks. Part I, published in 1985, contains an account of Chapters 1-9 in the second notebook as well as a description of Ramanujan's quarterly reports. In this volume, we examine Chapters 10-15 in Ramanujan's second notebook. If a result is known, we provide references in the literature where proofs may be found; if a result is not known, we attempt to prove it. Not only are the results fascinating, but, for the most part, Ramanujan's methods remain a mystery. Much work still needs to be done. We hope readers will strive to discover Ramanujan's thoughts and further develop his beautiful ideas.

2. Berndt, B. C. (1991) Ramanujan{\textquoteright}s Notebooks: Part III. Springer, Germany, 9780387975030, English (US), 10.1007/978-1-4612-0965-2, Identity, theorem, proof, function, equation, During the time period between 1903 and 1914, Ramanujan worked in almost complete isolation in India. Throughout these years, he recorded his mathematical results without proofs in notebooks. Upon Ramanujan's death in 1920, G.H. Hardy strongly urged that Ramanujan's notebooks be published and edited. The English mathematicians G.N. Watson and B.M. Wilson began this task in 1929, but although they devoted nearly ten years to the project, the work was never completed. In 1957, the Tata Institute of Fundamental Research in Bombay published a photostat edition of the notebooks, but no editing was undertaken. In 1977, Berndt began the tasks of editing Ramanujan's notebooks. Proofs are provided to theorems not yet proven in previous literature, and many results are so startling and different that there are no results akin to them in the literature.

3. Berndt, B. C. (1994) Ramanujan's notebooks. Part IV. Springer, Germany, 0-387-94109-6, English (US), 10.1007/978-1-4612-0879-2

4. Berndt, B. C. (1998) Ramanujan's notebooks. Part V. Springer, Germany, 0-387-94941-0, English (US), 10.1007/978-1-4612-1624-7

5. Ramanujan, S. (1957) Notebooks (2 volumes). Tata Institute of Fundamental Research, Bombay

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