Definition

Let \(f(x)\) be a formal power series defined as

\[f(x)=\sum_{n=0}^{+\infty}a_n\,x^n\,.\]

If \(N(x)\) and \(D(x)\,\) are polynomials of degree less or equal than \(p\) and \(q\,,\) respectively, with \(p\,,q\in \mathbb{N}_0\,,\) such that, \(D(0)\neq0\) and

\[f(x) - \frac{N(x)}{D(x)} = \mathcal{O}(x^{p+q+1})\]

then, the rational function \(N(x)/D(x)\) is a Padé approximant of \(f(x)\,.\) It can be denoted by \([p/q]_f(x)\,\) or \([p/q](x)\,\) or simply \([p/q]\,.\)

To ensure that the rational function \([p/q](x)\,\) doesn’t vanish at \(x=0\,,\) it is required that \(D(0)\neq0\,.\) To overcome this problem is assumed \(D(0)=1\,\) [1]. Then, using the first \(p+q+1\) coefficients of \(f(x)\) we only need to calculate \(p+q+1\) coefficients of \([p/q](x)\,.\)

[1] Brezinski, C. (1983). Outlines of Pade Approximation. In: Werner H., Wuytack L., Ng E., Bunger H.J. (eds) Computational Aspects of Complex Analysis. NATO Advanced Study Institutes Series (Series C | Mathematical and Physical Sciences), volume 102. Springer, Dordrecht.}