Cosmological Applications of Pade Approximant

As is well known, in mathematics, any function could be approximated by the Pad\'e approximant. The Pad\'e approximant is the best approximation of a function by a rational function of given order. In fact, the Pad\'e approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. In the present work, we consider the Pad\'e approximant in two issues. First, we obtain the analytical approximation of the luminosity distance for the flat XCDM model, and find that the relative error is fairly small. Second, we propose several parameterizations for the equation-of-state parameter (EoS) of dark energy based on the Pad\'e approximant. They are well motivated from the mathematical and physical points of view. We confront these EoS parameterizations with the latest observational data, and find that they can work well. In these practices, we show that the Pad\'e approximant could be an useful tool in cosmology, and it deserves further investigation.