High-Order Parametrization of the Hypergeometric-Meijer Approximants Abouzeid M. Shalaby

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High-Order Parametrization of the Hypergeometric-Meijer

Approximants

Abouzeid M. Shalaby∗

Physics Program, Department of Mathematics,

Statistics and Physics, College of Arts and Sciences,

Qatar University, P.O box 2713, Doha, Qatar

In previous articles, we showed that, based on large-order asymptotic behavior,

one can approximate a divergent series via the parametrization of a speciﬁc hyper-

geometric approximant. The analytical continuation is then carried out through a

Mellin-Barnes integral representation of the hypergeometric approximant or equiv-

alently using an equivalent form of the Meijer G-Function. The parametrization

process involves the solution of a non-linear set of coupled equations which is hard

to achieve (might be impossible) for high orders using normal PCs. In this work, we

extend the approximation algorithm to accommodate any order (high or low) of the

given series in a short time. The extension also allows us 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 x4anharmonic oscillator with and with-

out the non-perturbative information. We also considered the available 20 orders

for the ground sate energy of the PT −symmetric ix3anharmonic 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 describ-

ing strong-coupling and large-order asymptotic behaviors. The employment of the

non-perturbative data accelerated the convergence very clearly. The High tempera-

ture expansion for the susceptibility within the SQ lattice has been also considered

and led to accurate prediction for the critical exponent and critical temperature.

PACS numbers: 02.30.Lt,64.70.Tg,11.10.Kk

Keywords: Hypergeometric Approximants, High Temperature expansion, PT -symmetry

arXiv:2210.04575v1 [hep-th] 10 Oct 2022

I. INTRODUCTION

Frequently in physics, one is confronted by the existence of divergent perturbation series.

This can exist in more than one type of series behavior. There exists divergent series with

zero radius of convergence where perturbation fails to give reliable results for the whole

complex plane of the perturbation parameter. Another type is a series with ﬁnite radius of

convergence but the region of interest is outside the disk of convergence like critical region of

high temperature expansion. Examples in physics for the ﬁrst type include (but not limited

to) the expansion of physical quantities within the x4anharmonic oscillator, Ising model,

the φ4scalar ﬁeld theory and QED. To draw reliable results from such series one can apply

resummation techniques like Borel resummation [1–4], Pad´eapproximants [5–7] as well as

variational methods [1]. Recently, a hypergeometric-Borel technique [8] has also been applied

to resum divergent perturbation series . The hypergeometric approximants pFp−1[9,10]

have been also shown to produce good approximations for a divergent series with zero-radius

of convergence. However, the approximants pFp−1have a series expansion with ﬁnite-radius

of convergence and thus when used to approximate divergent series with zero-radius of

convergence they show some shortcomings [8,11,12]. In Refs.[13–15], we introduced what we

call it the hypergeometric-Meijer approximation algorithm. This algorithm can approximate

diﬀerent types of series based on the divergence manifestation. In fact, the type of divergence

is manifested in the growth factor of the series coeﬃcients at large orders. A series with ﬁnite

radius of convergence has 0! growth factor while the zero radius of convergence ones can have

n!,(2n)!, . . . growth factors. Our algorithm can treat such types of series and in fact it has

the same spirit of the hypergeometric approximant introduced by Mera et.al in Ref. [9] but

in a way that respects the analytic properties of the given-series and is able to accommodate

all known non-perurbative data associated with the given series. The employment of the

non-perturbative data is known to accelerate the convergence of resummation techniques

[1] and it has been shown in our previous work that it is accelerating the convergence of

the hypergeometric approximants as well. Our algorithm has been shown to give excellent

results for the approximation of diﬀerent divergent perturbation series [13–15].

The hypergeometric-Meijer algorithm is pretty simple (but accurate) and has two main

steps:

∗amshalab@qu.edu.qa

1. Approximating the given series with a hypergeometric series pFqthat can be parametrized

to reproduce all the known information about the original series.

2. The parametrized hypergeometric series is then analytically continued using its integral

representation in the form of a Mellin-Barnes integral or equivalently in terms of a

Meijer G function where [16]:

pFq(a1, ...ap;b1....bq;z) = Qq

k=1 Γ (bk)

k=1 Γ (ak)G1,p

p,q+11−a1,...,1−ap

0,1−b1,...,1−bqz,(1)

and

Gm,n

p,q c1,...,cp

d1,...,dqz=1

2πi ZCQn

k=1 Γ (s−ck+ 1) Qm

k=1 Γ (dk−s)

k=n+1 Γ (−s+ck)Qq

k=m+1 Γ (s−dk+ 1)zsds. (2)

For the hypergeometric approximants of interest pFq(a1, ...ap;b1....bq;z), where p=q+ 1

and p=q+ 2, the above integral representation is known to converge [13,16]. To show how

one can choose the suitable hypergeometric approximant for a given series, assume we are

given a series up to some order nfor a physical quantity Q(z) in the form:

Q(z)≈

cizi.

Based on its large-order behavior, a hypergeometric function pFq, with a constraint on

L=p−q, can be parametrized to accommodate all known information given for the series

under consideration. For a given series, one might know the ﬁrst nterms, the large-order

asymptotic behavior of the series and its asymptotic strong-coupling behavior. It is the large-

order behavior that determines the constraint on L. For instance, if the given series has a

ﬁnite radius of convergence then for large i,cibehaves as σiib. The radius of convergence R

is then 1

σ. In this case, the suitable hypergeometric approximants are pFqwith L= 1. In

case the series has a zero-radius of convergence with an asymptotic large-order behavior of

the form i!σiib, then the suitable approximants are pFq, with L= 2 and so on [13–15,17].

The hypergeometric approximant pFq(a1, a2,........ap;b1, b2, ....bq;σz) has the series expan-

sion:

pFq(a1, a2,.... ap;b1, b2.... bq,;σx) =

∞

n=0

Γ(a1+n)

Γ(a1)

Γ(a2+n)

Γ(a2).............Γ(ap+n)

Γ(a2)

n!Γ(b1+n)

Γ(b1)

Γ(b2+n)

Γ(b2).......Γ(bq+n)

Γ(bq)

(σx)n.(3)

For L= 1,2 , we have shown that it can be parametrized to produce the asymptotic large

order behavior [13–15,17] such that:

i=1

ai−

i=1

bi−L=b. (4)

Also, the numerator parameters (−ai) are representing the strong coupling parameters of

the given series [13]. However, technical problems in the calculation arise for ﬁnite values of

zwhen the diﬀerence ak−ajis an integer [18]. Accordingly, as we will explain later when we

impose the strong-coupling parameters into the approximants to accelerate the convergence,

it is more safer to employ the ﬁrst ai's with the diﬀerence ak−ajis not an integer to avoid

singularities in the calculations.

Let us now show how to use the hypergeometric approximants to approximate a given

series. For simplicity, assume ﬁrst that we have only the ﬁrst ﬁve orders of the perturbation

series:

Q(z)≈

cizi,

The weak-coupling parametrization assumes that we know the values of c0, c1, c2, c3, c4and

c5but the non-perturbative parameters are not known. The ratio test can tel us about the

radius of convergence of the given series. If the given series is known to have a zero radius of

convergence with coeﬃcients cibehave like i!σiibfor large i, then the suitable approximant

Q(z)≈c0 3F1(a1, a2,a3,;b1;σz).

The approximant 3F1(a1, a2,a3,;b1;σz) has ﬁve parameters, namely a1, a2,a3,b1and σto be

determined. Matching coeﬃcients of same order of zin the given series and the series

expansion of c0 3F1we get;

a1a2a3

σ=c1

a1a2a3(a1+ 1) (a2+ 1) (a3+ 1)

2!b1(b1+ 1) σ2=c2

a1a2a3(a1+ 1) (a2+ 1) (a3+ 1) (a1+ 2) (a2+ 2) (a3+ 2)

3!b1(b1+ 1) (b1+ 2) σ3=c3

a1a2a3(a1+ 1) (a2+ 1) (a3+ 1) ...... (a1+ 3) (a2+ 3) (a3+ 3)

4!b1(b1+ 1) .... (b1+ 3) σ3=c4(5)

a1a2a3(a1+ 1) (a2+ 1) (a3+ 1) ...... (a1+ 4) (a2+ 4) (a3+ 4)

5!b1(b1+ 1) .... (b1+ 4) σ3=c5

This a set of non-linear algebraic equations can be solved for the ﬁve unknowns a1, a2,a3, b1, σ.

The degree of non-linearity can be lowered by generating the ratio Rn=cn

cn−1and match it

by the corresponding ratio gnfrom the series expansion of the hypergeometric approximant

where

pFq(a1, a2,........ap;b1, b2, ....bq;σz) =

∞

n=0

hnzn,

and

gn=hn

hn−1

i=1

(ai+n−1)

j=1

(bj+n−1)

σ. (6)

The set of equations Rn=gnis still non-linear and in going to higher orders will make it very

hard and might be impossible to solve it in a practical time using normal computers. In fact,

this represents a major obstacle that prevents the current versions of hypergeometric-Meijer

algorithm from tackling the approximation of a divergent perturbation series with relatively

high orders used as input. In literature, one can ﬁnd perturbation series obtained up to

a relatively high order like the high-temperature expansion of Ising like models [19,20]

for which the the hypergeometric approximants pFp−1(a1, a2,........ap;b1, b2, ....bp−1;z) oﬀer

a good approximation for the given series. Likewise, the ground state energy for both

hermitian x4[21] and the non-Hermitian ix3[22] anhrmoinic oscillators are known up to

high orders and thus the parametrization of the hypergeometric approximants that can

accommodate information from the known orders is necessary. In this work, we introduce

a simple algorithm to get an equivalent (order by order) set of linear equations that can be

solved easily using a normal PC and for short time. Note that in Ref. [8], Mera et. al used

the hypergeometric approximants pFp−1(a1, a2,........ap;b1, b2, ....bp−1;z) to approximate the

Borel series obtained by Borel transforming the given perturbation series. They introduced

the ansatz (Eq.(5) in the same reference) to approximate the ratio gnfor pFp−1. Although

the important idea of getting a linear set of equations followed by Mera et.al is similar to

what we will follow in our Hypergeometric-Meijer algorithm, in our work however, we do

not use any ansatz and shall try to get a linear set of equations not only for pFp−1but

for any hypergeometric approximant pFq. We need to assert that we shall get a set of

linear equations that is completely equivalent (order by order) to the original set without

any approximation. Moreover, the linear set in our work is able to accommodate the non-

perturbative data as well. In the following sections we apply the algorithm for diﬀerent

problems and for diﬀerent type of series. The application of the algorithm will address ﬁrst

the weak-coupling parametrization and then will deal with cases of a mixture of information

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High-OrderParametrizationoftheHypergeometric-MeijerApproximantsAbouzeidM.ShalabyPhysicsProgram,DepartmentofMathematics,StatisticsandPhysics,CollegeofArtsandSciences,QatarUniversity,P.Obox2713,Doha,QatarInpreviousarticles,weshowedthat,basedonlarge-orderasymptoticbehavior,onecanapproximateadivergents...

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High-Order Parametrization of the Hypergeometric-Meijer Approximants Abouzeid M. Shalaby

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