Hypergeometric Series and Approximations of Mathematical Constants

I will discuss some identities for generalized hypergeometric series that were discovered quite recently in connection with rational approximations to π, π2 and π4. A curious thing is that most of these identities have “automatic” proofs (using creative telescoping), and a problem is to provide “human” proofs by means of classical hypergeometric summation and transformation formulas. Let me say from the beginning that the subject of my lecture is a brief exposition of two topics in number theory, about rational approximations to numbers of the form √ d/π and √ d/π, and about Apéry-like rational approximations to the number ζ(4) = π/90. These approximations are deeply related to certain unusual transformations of generalized hypergeometric series. Some of these transformations can be shown by means of creative telescoping due to Gosper–Zeilberger, but this does not provide us a way to deduce a general form for such transformations which is really needed to do some new results in number theory. I address the problems of providing human proofs for the transformations and their generalizations to specialists in special functions well represented at this conference. 1. Guillera’s generalization of Ramanujan’s series for 1/π In 1914 S. Ramanujan recorded a list of 17 series for 1/π, from which we indicate the simplest one (1) ∞ ∑ n=0 ( 1 2 ) 3 n n!3 (4n+ 1) · (−1) = 2 π and also two quite impressive examples ∞ ∑ n=0 ( 1 4 )n( 1 2 )n( 3 4 )n n!3 (21460n+ 1123) · (−1) n 8822n+1 = 4 π , (2) ∞ ∑ n=0 ( 14 )n( 1 2 )n( 3 4 )n n!3 (26390n+ 1103) · 1 994n+2 = 1 2π √ 2 (3) which produce rapidly converging (rational) approximations to π. Here (a)n = Γ(a+ n) Γ(a) = { a(a+ 1) · · · (a+ n− 1) for n ≥ 1, 1 for n = 0, Date: July 5, 2007. A talk at the 9th Conference on Orthogonal Polynomials, Special Functions and Applications (Marseille, France, July 2–6, 2007). 1