Abstract
We derive many new formulas for approximation of pi. The formulas make use of a sequence of iteration functions called the Basic Family; a nontrivial determinantal generalization of Taylor’s theorem; other ingredients; as well as several new results presented in the present paper. In one scheme, one evaluates members of the Basic Family, for an appropriately selected function, all at the same input. This scheme generates almost a fixed and pre-selected number of digits in each successive evaluation. The computation amounts to the evaluation of a recursive formula and is equivalent to the computation of special Toeplitz matrix determinants. The approximations of pi obtained via this scheme are within simple algebraic extensions of the rational field. In a second scheme, the fixed-point iteration is applied to any fixed member of the Basic Family, while selecting an appropriate function. In this scheme we prove high-order of convergence from the initial point. We report on some preliminary computational results obtained via Maple. Analogous formulas can be used to approximate other transcendental numbers. For instance, we also give a formula for approximation of e.