Ramanujan's Series for 1/π: A Survey

Abstract

When we pause to reflect on Ramanujan's life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathematics be brought to posterity. One of these was V. Ramaswamy Aiyer's founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the next necessary episode, namely, Ramanujan's meeting with Ramaswamy Aiyer at his office in Tirtukkoilur in 1910, would also have not taken place. Ramanujan had carried with him one of his notebooks, and Ramaswamy Aiyer not only recognized the creative spirit that produced its contents, but he also had the wisdom to contact others, such as R. Ramachandra Rao, in order to bring Ramanujan's mathematics to others for appreciation and support. The large mathematical community that has thrived on Ramanujan's discoveries for nearly a century owes a huge debt to V. Ramaswamy Aiyer. * This paper was originally solicited by the Editor of Mathematics Student to commemorate the founding of the Indian Mathematical Society in its centennial year. Mathematics Student is one of the two official journals published by the Indian Mathematical Society, with the other being the Journal of the Indian Mathematical Society. The authors thank the Editor of Mathematics Student for permission to reprint the article in this MONTHLY with minor changes from the original.

Mosaic Supercongruences of Ramanujan Type

“…An excellent survey on Ramanujan-type series is [2] and a beautiful survey on recent advances is in [10]. There are many examples of convergent Ramanujan-type and Ramanujan-Sato-type series in the literature, the most spectacular are of simple and very fast series.…”

“…They are of the following form During a long time these Ramanujan's series for 1/π were almost ignored but since 1987 the interest of mathematicians in them was great. The 17 formulas as well as many other series of the same type are already proved rigorously [2]. Naturally, they are called Ramanujan-type series.…”

Арифметические Гипергеометрические Ряды

“…(см., например, [111]) могут быть получены единообразно благодаря доступно-му модулярному и гипергеометрическому аппаратам [30], [16], [181]. Подобная формула…”

“…В настоящее время уже практически нет ничего загадочного в классических формулах Рамануджана для представления 1/π, примеры которых -формулы (4)-(6) и их многочис-ленные обобщения. По этому вопросу мы рекомендуем читателю ознакомиться с монографией [30] и недавно опубликованными обзорами [16], [181]. В от-ношении данных формул примечательным является не только явное присут-ствие гипергеометрического ряда в левой части, но также наличие чисто ги-пергеометрического аппарата [51], [65], [67], позволяющего доказывать неко-торые из таких представлений.…”

New hypergeometric-like series for $1/\pi^{2}$ arising from Ramanujan’s theory of elliptic functions to alternative base 3

“…These authors and several other authors in the past two decades have found many new series for 1/π as well as for 1/π 2 . We refer to [4] for a recent survey on Ramanujan's series for 1/π.…”