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