On computing the coefficients of bivariate holonomic formal series

In this work, we study the problem of computing the coefficients of holonomic formal series in two commuting variables. Given a formal series φ(x,y)=∑n,k⩾0cnkxnyk specified by a holonomic system ∑j=0d1pj(x,y)∂xjφ=0 and ∑j=0d2qj(x,y)∂yjφ=0, with a suitable finite set of initial conditions {[xayb]φ(x,y)}, we show that the coefficient [xiyj]φ(x,y) can be computed in time O(i+j) under the uniform cost criterion.