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A general theorem is stated that unifies 93 rational Ramanujan-type series for 1/π, 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, thereby giving another 93 series, the majority of which are new.

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.

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Abstract.A new infinite product t n was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan's assertions about t n by establishing new connections between the modular jinvariant and Ramanujan's cubic theory of elliptic functions to alternative bases. We also show that for certain integers n, t n generates the Hilbert class field of Q( √ −n). This shows that t n is a new class invariant according to H. Weber's definition of class invariants.

We resolve a family of recently observed identities involving 1/π using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates a generating function of the Legendre polynomials to a product of two Gaussian hypergeometric functions. Using our methods, we also derive some new Ramanujan-type series.

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In this article, we investigate congruences satisfied by Apéry-like numbers.