Series and iterations for 1/π

Rational analogues of Ramanujan's series for 1/π

“…A different proof is outlined in the expository article by W. Duke [34]. Proofs of the identities for level 6 in cases B and C that use series manipulations have been given in [30]. The other identities can be proved in the same way, or by using the ideas in [20] or [34].…”

“…The study of series for 1/π has a long and distinguished history, and the reader is referred to [3] or [17]. However it would be remiss not to mention [4, 9, 11, 13-15, 21, 22, 26, 39] in connection with levels 1-4; [30] and [42] for levels 5 and 6; and [19,23,24,41] for level 6. We end this introduction by mentioning that there are some series for 1/π that do not involve a weight one modular form such as (5) but are instead deduced directly from weight two modular forms without the use of a Clausen-type identity. See [23], [31] and [46] for some series for level 10 that were inspired by an example of Y. Yang, and see [32] for some series for levels 7 and 18.…”

Congruences modulo powers of 2 for a certain partition function

“…Interchanging the order of summation and using properties (5), (6), and (11) of the numbers m and b to extend the sums to all positive integers, we get ∞ n=0 c 2 2j n + 1 3 2 2j +1 + 1 q n+1…”

New analogues of Ramanujan's partition identities

“…(See also [10].) The existence of identities (1.4), (1.5), (1.7) and (1.8) can all be explained by the theory of modular functions.…”