Fast computation of the N-th term of a q-holonomic sequence and applications

In 1977, Strassen invented a famous baby-step/giant-step algorithm that computes the factorial ! in arithmetic complexity quasi-linear in . In 1988, the Chudnovsky brothers generalized Strassen's algorithm to the computation of the -th term of any holonomic sequence in essentially the same arithmetic complexity. We design -analogues of these algorithms. We first extend Strassen's algorithm to the computation of the -factorial of , then Chudnovskys' algorithm to the computation of the -th term of any -holonomic sequence. Both algorithms work in arithmetic complexity quasi-linear in ; surprisingly, they are simpler than their analogues in the holonomic case. We provide a detailed cost analysis, in both arithmetic and bit complexity models. Moreover, we describe various algorithmic consequences, including the acceleration of polynomial and rational solving of linear -differential equations, and the fast evaluation of large classes of polynomials, including a family recently considered by Nogneng and Schost.