Entire hypergeometric approximants for the ground state energy perturbation series of the quartic, sextic and octic anharmonic oscillators

Abstract

In resumming a divergent series, one is frequently confronted with situations where predictions from traditional resummation techniques might not accurate when considering strong-coupling values. Besides, in literature, we barely can find a resummation algorithm for which the convergence is faster for strongly-divergent ( Gevrey−k , k>1) series than a divergent one ( Gevrey−1 ). In this work, we use appropriate hypergeometric functions to approximate a divergent series ( n! large order growth factor) and strongly-divergent series ( (2n)! and (3n)! large order growth factors). We found that the convergence of the predicted non-perturbative parameters to their exact values is faster for the challenging Gevrey−3 than the Gevrey−2 series which in turn is faster than the divergent Gevrey−1 series. To explain such interesting feature, we suggest a type of analytic continuation where the parametrized divergent hypergeometric series is represented as a finite sum over entire hypergeometric functions. The algorithm is applied to sum the weak-coupling series of the ground state energy of the quartic (divergent), sextic (Gevrey−2) and Octic (Gevrey−3) anharmonic oscillators. Accurate results are obtained for the ground state energy as well as the predicted non-perturbative parameters. While traditional resummation techniques fail to give reliable results for large coupling values, our algorithm gives the expected limit as the coupling goes to infinity.