Hypergeometric Gevrey-0 approximation for the Gevrey-k divergent series with application to eight-loop renormalization group functions of the O(N)-symmetric field model

Abstract Mera et al. (Phys Rev Lett 115: 143001, 2015) discovered that the hypergeometric function ₂F₁ (a₁, a₂;b₁; g) 2 F 1 (a 1, a 2 ; b 1 ; ω g) can serve as an accurate approximant for a divergent Gevrey-1 type of series with an asymptotic large-order behavior of the form n! nᵇ ⁿ n ! n b σ n. What is strange about this approximant is that it has a series expansion with the wrong large-order behavior (Gevrey-0 type). In this work, we extend this discovery to Gevrey-k series where we show that the hypergeometric approximants and its extension to the generalized hypergeometric approximants are not only able to approximate divergent (Gevrey-1) series but also able to approximate strongly-divergent series of Gevrey-k type with k=2, 3, k = 2, 3, …. Moreover, we show that these hypergeometric approximants are able to predict accurate results for the non-perturbative strong-coupling and large-order parameters from weak-coupling data as input. Examples studied here are the ground-state energy for the xⁿ x n anharmonic oscillators. The hypergeometric approximants are also used to approximate the recent eight-loop series (g -expansion) of the renormalization group functions for the O (N) -symmetric ⁴ ϕ 4 scalar field model. Form these functions for N=0, 1, 2 N = 0, 1, 2, and 3, critical exponents are extracted which are very competitive to results from more sophisticated approximation techniques.