A new infinite product tn was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujanâ€™s assertions about tn by establishing new connections between theâ€¦ (More)

We resolve a family of recently observed identities involving 1/Ï€ using the theory of modular forms and hypergeometric series. In particular, we resort to a formula of Brafman which relates aâ€¦ (More)

In his lost notebook, Ramanujan offers several results related to the crank, the existence of which was first conjectured by F. J. Dyson and later established by G.E.Andrews and F.G. Garvan. Using anâ€¦ (More)

where K' = K(k') and k' = y/l-k is the complementary modulus. Thus, an evaluation of any one of the functions <p, 2F\, or K yields an evaluation of the other two functions. However, such evaluationsâ€¦ (More)

When we pause to reflect on Ramanujanâ€™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 wasâ€¦ (More)

In this paper, we use the explicit Shimura Reciprocity Law to compute the cubic singular moduli Î± * n , which are used in the constructions of new rapidly convergent series for 1/Ï€. We also completeâ€¦ (More)

A survey of Ramanujanâ€™s work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujanâ€¦ (More)

In his lost notebook, Ramanujan defined a parameter Î»n by a certain quotient of Dedekind eta-functions at the argument q = exp(âˆ’Ï€âˆšn/3). He then recorded a table of several values of Î»n. To proveâ€¦ (More)