Calculation of hypergeometric series with quasi-linear time and linear space complexity

A simple for practical implementation algorithm with the time complexity ${\mathsf O}(M(n)\log(n)^2)$ and space complexity ${\mathsf O}(n)$ for the evaluation of hypergeometric series with rational coefficients on the Schönhage machine is constructed (here $M(n)$ is the complexity of integer multiplication). It is shown that this algorithm is suitable in practical informatics for constructive analogues of often used constants of analysis.