A simple algorithm for the evaluation of the hypergeometric series using quasi-linear time and linear space

A simple algorithm with time complexity O(M(n)log(n) 2 ) and space complexity O(n) for the evaluation of the hypergeometric series with rational coefficients is con- structed (M(n) being 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. Introduction. In this paper we construct an algorithm for the calculation of the approximate values of the hypergeometric series with rational coefficients whose imple- mentation is simple and which is quasi-linear in time and linear in space on the machine Schonhage (1). Such series are used for the calculation of some mathematical constants of analysis and of the values of elementary functions at rational points. Sch(FQLIN − TIME//LIN − SPACE) will be used to denote the class of algo- rithms which are computable on Schonhage and are quasi-linear in time and linear in space. The main feature of Schonhage is its ability to execute recursive calls of proce- dures. Quasi-linear means that the complexity function is bounded by O(nlog(n) k ) for some k. The main advantage of algorithms based on series expansions is the relative simplic- ity of both the algorithms and the analysis of their computational complexity. Besides we can compute all the most commonly used constants of analysis using series expan- sions. For calculations with a small number of digits after the binary (or decimal) point, series are more efficient than other methods because of the small constants in estima- tions of their computational complexity. Therefore such algorithms are important in computer science for practical applications. It is known (2) that linearly convergent hypergeometric series with rational coef- ficients can be calculated using the binary splitting method with time complexity O(M(n)(log(n)) 2 ) and space complexity O(nlog(n)) (where M(n) denotes the com- plexity of multiplication of n-bit integers). In recent publications, for example (3), algorithms based on a modified binary splitting method for the evaluation of linearly convergent hypergeometric series with time complexity O(M(n)(log(n)) 2 ) and space complexity O(n) are described.