Calculating the continued fraction coefficients of a sub-diagonal Pad\'e approximant at arbitrary order

Understand

The inspiral of two compact objects in gravitational wave astronomy is described by a post-Newtonian expansion in powers of $(v/c)$.

• In most cases, it is believed that the post-Newtonian expansion is asymptotically divergent.
• A standard technique for accelerating the convergence of a power series is to re-sum the series by means of a rational polynomial called a Pad\'e approximation.
• If we liken this approximation to a matrix, the best convergence is achieved by staying close to a diagonal Pad\'e approximation.