A Class of Conjectured Series Representations for 1/π

Using the second conjecture in the paper [10] of the author and inspired by the theory of modular functions we find a method which allows us to obtain explicit formulae, involving η or θ functions, for the parameters of a class of series for 1/π. As in [10], the series considered in this paper include Ramanujan’s series as well as those associated with the Domb numbers and Apéry numbers. 1 A special type of recurrence The sequence of integer numbers Bn = (2n)! n!6 , (1) satisfies the following recurrence nBn − 8(2n− 1)Bn−1 = 0. Other sequences of integer numbers satisfying a first order recurrence whose coefficients are third degree polynomials: Bn = (4n)! n!4 , (2) Bn = (2n)!(3n)! n!5 , (3) and Bn = (6n)! (3n)!n!3 , (4) satisfy the recursions nBn − 8(2n− 1)(4n− 3)(4n− 1)Bn−1 = 0, 1 nBn − 6(2n− 1)(3n− 2)(3n− 1)Bn−1 = 0 and nBn − 24(2n− 1)(6n− 5)(6n− 1)Bn−1 = 0, respectively. Examples of sequences of integers which satisfy a second order recurrence with third degree polynomials as coefficients are [1]: The sequence of Domb numbers [5]