High-order parametrization of the hypergeometric-Meijer approximants

Abstract

In this work, we introduce an extension to the hypergeometric algorithm we developed before for the resummation of divergent series. The extension overcome the time-consuming problem we face in the parametrization process of the hypergeometric approximants. In the previous version, the parametrization process results in a set of non-linear equations which is hard to solve (might be impossible) for relatively high orders (greater than six) using normal PCs with Mathematica software for instance. To solve this problem, we formulate an equivalent (order by order) linear set of equations which is easy to solve in an appropriate time using normal PCs. We also show that such extension of the hypergeometric resummation algorithm is able to employ non-perturbative information like strong-coupling and large-order asymptotic data which are always used to accelerate the convergence. We applied the algorithm for different orders (up to O(29)) of the ground state energy of the x4 anharmonic oscillator with and without the non-perturbative information. We also considered the available 20 orders for the ground state energy of the PT−symmetric ix3 anharmonic oscillator as well as the given 20 orders of its strong-coupling expansion or equivalently the Yang–Lee model. For high order weak-coupling parametrization, accurate results have been obtained for the ground state energy and the non-perturbative parameters describing strong-coupling and large-order asymptotic behaviors. The employment of the non-perturbative data accelerated the convergence very clearly. The High temperature expansion for the susceptibility within the SQ lattice has been also considered and led to accurate prediction for the critical exponent and critical temperature.