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

AbstractMera et al. (Phys Rev Lett 115:143001, 2015) discovered that the hypergeometric function $${}_2F_1(a_1,a_2;b_1; \omega g)$$ 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 $$n! n^b \sigma ^n$$ 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 $$k=2,3,\ldots$$ 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 $$x^n$$ 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 $$\phi ^4$$ 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.