Fast Summation of Power Series with Coefficients Analytic at Infinity

Abstract

We propose a fast summation algorithm for slowly convergent power series of the form ∑ _{j=j 0} ^∞ z ^jφ _j j ^μ∏ _i=1 ^s(j+α _i )^−ν _i , where μ∈R, ν _i ≥0 and α _i ∈C, 1≤i≤s, are known parameters, and φ _j =φ(j), φ being a given real or complex function, analytic at infinity. Such series embody many cases treated by specific methods in the recent literature on acceleration. Our approach rests on explicit asymptotic summation, started from the efficient numerical computation of the Laurent coefficients of φ. The effectiveness of the resulting method, termed ASM (Asymptotic Summation Method), is shown by several numerical tests.