Elliptic integrals and Ramanujan-type series for 1/π associated with Γ0(N), where N is a product of two small primes

Abstract

In the lost notebook Ramanujan stated results for elliptic integrals associated with Γ 0 ( N ) for N = 5 , 7, 10, 14, 15 and 35. But he did not record an integral corresponding to N = 21 despite having worked out all of the relevant P – Q modular identities. Possibly, the reason for the omission is that the corresponding differentiation formula is more complicated. In this work, it is shown that under an appropriate change of variables, a particularly simple differentiation formula for N = 21 can be obtained. It turns out to be just one piece of a larger theory, and we show how the level 21 theory can be extended to produce Ramanujan-type series for 1 / π . We also outline how a similar procedure can be used to produce the corresponding results for N = 22 , 33 and 35.