Linear Recurrences, Linear Diierential Equations, and Fast Computation 1. Classical Algorithms concerning Formal Power Series

summary by Philippe Dumas] Linear recurrences and linear diierential equations with polynomial coeecients provide a nite representation of special functions or special sequences. Many algorithms are at our disposal; some give a way to automate the computation of recurrences or diierential equations; some provide solutions to recurrences or diierential equations; and some give the asymptotic behaviour of these solutions, directly from the recurrence or diierential equation. All of this provides a method to eeciently compute special functions and special sequences. In the sequel, we use the ring A x]] of formal power series F(x) = +1 X n=0 f n x n with coeecients f n in a commutative ring A ; this ring is assumed to contain the eld Q of rational numbers, even though it is possible to consider a more general situation. Practically, one deals with truncated series F(x) = N X n=0 f n x n + O(x N+1); that is to say essentially polynomials. It must be noted that there exist lazy algorithms to deal with truncated series of arbitrary order, but their cost is generally excessive. We indicate how to deal with basic operations 7, Chap. 4].