Ramanujan-type formulae and irrationality measures of some multiples of $\pi$

An explicit construction of simultaneous Padé approximations for generalized hypergeometric series and formulae for the quantities π √ d , d∈{1, 2, 3, 10005}, in terms of these series are used for estimates of irrationality measures of these multiples of π. Other possible applications are also discussed. Bibliography: 14 titles. Introduction An important role in the history of the Archimedes constant π is played by formulae allowing one to calculate it with high accuracy (these days one speaks about billions of decimals). One class of these formulae are representations obtained by Ramanujan in 1914 [1], among which we must point out first of all the following two examples: ∞ ∑ ν=0 (1/4)ν(1/2)ν(3/4)ν ν!3 (21460ν + 1123) · (−1) ν 8822ν+1 = 4 π , (1) ∞ ∑ ν=0 (1/4)ν(1/2)ν(3/4)ν ν!3 (26390ν + 1103) · 1 994ν+2 = 1 2π √ 2 ; (2) As usual, (a)ν = Γ(a+ν)/Γ(a) = a(a+1) · · · (a+ν−1) for ν 1 and (a)0 = 1 is the Pochhammer symbol (the shifted factorial); here and throughout, ‘empty’ products are set equal to 1. These formulae have only recently been rigorously substantiated and until now new formulae of Ramanujan type have arisen in connection with modular parametrization of solutions of differential equations [2] and algorithms for hypergeometric series [3]. We present as examples two further formulae, which we shall use in the present paper: ∞ ∑ ν=0 (1/3)ν(1/2)ν(2/3)ν ν!3 (14151ν + 827) · (−1) ν 5002ν+1 = 3 √ 3 π , (3) ∞ ∑ ν=0 (1/6)ν(1/2)ν(5/6)ν ν!3 (545140134ν + 13591409) · (−1) ν 533603ν+2 = 3 2π √ 10005 ; (4) This research was carried out with the partial support of the Russian Foundation for Basic Research (grant no. 03-01-00359). AMS 2000 Mathematics Subject Classification. Primary 11J82, 41A2; Secondary 33C20.