Journal of Physics Communications

PAPER • OPEN ACCESS

A new method to sum divergent power series:
educated match
To cite this article: Gabriel Álvarez and Harris J Silverstone 2017 J. Phys. Commun. 1 025005

 

View the article online for updates and enhancements.

Related content
On the divergence of gradient expansions
for kinetic energy functionals in the
potential functional theory
Alexey Sergeev, Raka Jovanovic, Sabre
Kais et al.

-

Uniform asymptotic and JWKB expansions
for anharmonic oscillators
Gabriel Álvarez and Carmen Casares

-

Living without supersymmetry—the
conformal alternative and a dynamical
Higgs Boson
Philip D Mannheim

-

This content was downloaded from IP address 147.96.14.16 on 04/09/2018 at 18:21

https://doi.org/10.1088/2399-6528/aa8540
http://iopscience.iop.org/article/10.1088/1751-8113/49/28/285202
http://iopscience.iop.org/article/10.1088/1751-8113/49/28/285202
http://iopscience.iop.org/article/10.1088/1751-8113/49/28/285202
http://iopscience.iop.org/article/10.1088/0305-4470/33/13/304
http://iopscience.iop.org/article/10.1088/0305-4470/33/13/304
http://iopscience.iop.org/article/10.1088/1361-6471/aa888e
http://iopscience.iop.org/article/10.1088/1361-6471/aa888e
http://iopscience.iop.org/article/10.1088/1361-6471/aa888e


J. Phys. Commun. 1 (2017) 025005 https://doi.org/10.1088/2399-6528/aa8540

PAPER

A newmethod to sum divergent power series: educatedmatch

Gabriel Álvarez1 andHarris J Silverstone2,3

1 Departamento de Física Teórica II, Facultad deCiencias Físicas, UniversidadComplutense, E-28040,Madrid, Spain
2 Department of Chemistry, The JohnsHopkinsUniversity, 3400N.Charles Street, Baltimore,MD21218,United States of America
3 Author towhomany correspondence should be addressed

E-mail: galvarez@fis.ucm.es andhjsilverstone@jhu.edu

Keywords: summability, perturbation theory, Borel, Gevrey

Abstract
Wepresent amethod to sumBorel- andGevrey-summable asymptotic series bymatching the series to
be summedwith a linear combination of asymptotic series of known functions that themselves are
scaled versions of a single, appropriate, but otherwise unrestricted, functionΦ. Both the scaling and
linear coefficients are calculated fromPadé approximants of a series transformed from the original
series byΦ.We discuss in particular the case thatΦ is (essentially) a confluent hypergeometric
function, which includes as special cases the standard Borel–Padé andBorel–Leroy–Padémethods. A
particular advantage is themechanism to build knowledge about the summed function into the
approximants, extending their accuracy and range evenwhen only a few coefficients are available.
Several examples from field theory andRayleigh–Schrödinger perturbation theory illustrate the
method.

1. Introduction

Summation of divergent asymptotic expansions has led to a vast literature frombothmathematical and physical
points of view. Themathematical goal is often to assign a standard sum to a series whose coefficients satisfy
certain growth conditions andwhose sum satisfies certain conditions at infinity [1, 2]. The physical literature
focuses on awide range of specialized, computationalmethods. Especially since thework carried out in the
1970s on the coupling constant analyticity of anharmonic oscillators [3–5], two summationmethods have
become dominant: Padé approximants and Borel summation. Both have been found useful infields as diverse as
quantummechanics, statisticalmechanics, quantumfield theory, and string theory. Padé approximants are
most often directly used empirically (see, for example, the recent study on the existence of an ultraviolet zero for
the six-loop beta function of the 4

4lF theory [6]), and at timeswith new, alternative transformation procedures
[7]. Borel summability has been rigorously proved in several instances. The analytic continuation implicit in the
Borel summation process poses a practical problem that has been dealt with in essentially twoways: conformal
mappings [8–10], and Borel–Padé approximants. In the latter, the analytic continuation is again performed
empirically by Padé approximants of the Borel-transformed series [3, 11–13].Most recently,Mera et al [14, 15]
and Pedersen et al [16] have developed amethod that uses hypergeometric functions to sumperturbation theory
series using only a few terms.

Initiallymotivated in part by the papers ofMera et al, we present here a newmethod to build concise,
explicit, analytic approximants to the Borel orGevrey sumof an asymptotic power series. These approximants
match the series to be summedwith a linear combination of asymptotic series of known functions. The known
functions are scaled versions of a single functionΦ, and both the scaling and linear coefficients are readily
calculated fromPadé approximants of a transformed series determined by the original series and byΦ. IfΦ is
taken to be (essentially) a confluent hypergeometric function, the newmethod includes as special cases the
standard Borel–Padé andBorel–Leroy–Padé summationmethods. Evenmore important, prior additional (i.e.,
educated) knowledge about the summed function can be built into the approximants via the functionΦ,
sometimes dramatically extending the accuracy and range of the approximants. The ‘linear combination’here is
similar to the linear combination of the Janke–Kleinert resummation algorithm, which is described as

OPEN ACCESS

RECEIVED

19May 2017

ACCEPTED FOR PUBLICATION

9August 2017

PUBLISHED

26 September 2017

Original content from this
workmay be used under
the terms of the Creative
CommonsAttribution 3.0
licence.

Any further distribution of
this workmustmaintain
attribution to the
author(s) and the title of
thework, journal citation
andDOI.

© 2017TheAuthor(s). Published by IOPPublishing Ltd

https://doi.org/10.1088/2399-6528/aa8540
mailto:galvarez@fis.ucm.es
mailto:hjsilverstone@jhu.edu
http://crossmark.crossref.org/dialog/?doi=10.1088/2399-6528/aa8540&domain=pdf&date_stamp=2017-09-26
http://crossmark.crossref.org/dialog/?doi=10.1088/2399-6528/aa8540&domain=pdf&date_stamp=2017-09-26
http://creativecommons.org/licenses/by/3.0
http://creativecommons.org/licenses/by/3.0
http://creativecommons.org/licenses/by/3.0


‘re-expanding the asymptotic expansion in a complete set of basis functions’, andwhich ismathematically
equivalent to conformalmapping techniques [17]. Ourmethod, in contrast, is essentially linked to the theory of
Padé approximants.

2.Φ-Padé approximants

Our goal is to approximate the Borel sum zy ( ) of a divergent power series,

z d z , 1
k

k
k

0
åy ~
=

¥

( ) ( )

using any appropriate known function zF( )with its ownBorel-summable series,

z f z . 2
k

k
k

0
åF ~ -
=

¥

( ) ( ) ( )

Themethod is at the same time hidden in, and a generalization of, the Borel–Padé summationmethod [13],
whichwe briefly review.

Let us denote by P z Q zn n1- ( ) ( ) the n n1,-[ ]Padé approximant of the Borel transformof the series(1),

z
d

k
z , 3

k

k k
B

0
åy =
=

¥
ˆ ( )

!
( )

and let us assume thatQn(z)has only simple zeros. Partial fraction expansion,

P z

Q z

r

z z
, 4n

n j

n
j

j

1

1
å=

-
-

=

( )
( )

( )

and term-by-term integration lead to the standard Borel–Padé approximant zn nB, 1,y - ( )[ ] to zy ( ),

z
r

zt z
te d , 5n n

t

j

n
j

j
B, 1,

0 1
ò åy =

-
-

¥
-

=

( ) ( )[ ]

r

z
E z z , 6

j

n
j

j
j

1
Eulerå=

-
-

=

( ) ( )

wherewe define E zEuler ( ) by

E z
zt

t z E z
e

1
d e 1 , 7

t
z

Euler
0

1 1
1ò=

+
=

¥ -
-( ) ( ) ( )

andwhere E z11( ) is a standard version of the exponential integral (see chapter 5 of [18]).
The two points to note in this derivation are (i) that the E zEuler ( ) in equation (7) is precisely the Borel sumof

the factorially divergent Euler series [19] obtained by setting f kk = ! in equation (2), and (ii) that the asymptotic
expansion of zn nB, 1,y - ( )[ ] is identical to that of zy ( ) through order z n2 1- , i.e., that

r

z
E z z d z O z . 8

j

n
j

j
j

k

n

k
k n

1
Euler

0

2 1
2å å

-
- = +

= =

-

( ) ( ) ( )

In principle, the n2 parameters zj and rj could have been determined de nouveau from equation (8) by
substituting in it the Euler series and equating coefficients.

These observationsmotivate the following generalization of the Borel–Padé summationmethod.We define
the ‘Φ-transform’ of the series given in equation (1) by

z
d

f
z , 9

k

k

k

k

0
åy =F
=

¥
ˆ ( ) ( )

and the associated new approximants zn n, 1,yF - ( )[ ] to zy ( ) by

z
r

z
z z . 10n n

j

n
j

j
j, 1,

1
åy =

-
F -F -

=

( ) ( ) ( )[ ]

As a generalization of equation (8), the approximants zn n, 1,yF - ( )[ ] , which depend on n2 parameters
r z j n, , 1, ,j j = ¼( ), satisfy

r

z
z z z O zd , 11

j

n
j

j
j

k

n

k
k n

1 0

2 1
2å å

-
F - = +

= =

-

( ) ( ) ( )

2

J. Phys. Commun. 1 (2017) 025005 GÁlvarez andH J Silverstone



and therefore the rj and zj solve the n2 equations

r

z
f z d k n, 0, 1, , 2 1 . 12

j

n
j

j
k j

k
k

1
å

-
= = ¼ -

=

-( ) ( ) ( )

In practice, these parameters aremost easily calculated from the partial fraction expansion of the n n1,-[ ]
Padé approximant to zyFˆ ( ), i.e.,

P z

Q z

d

f
z O z

r

z z
. 13n

n k

n
k

k

k n

j

n
j

j

1

0

2 1
2

1
å å= + =

-
-

=

-

=

( )
( )

( ) ( )

In otherwords, the zj are the poles, for simplicity assumed to be simple, and the rj the residues, of the n n1,-[ ]
Padé approximant to zyFˆ ( ). Accordingly we call zn n, 1,yF - ( )[ ] the ‘ n n1,- F[ ] -Padé approximant’ to zy ( ).

The Borel–Padé approximant uses no information about the sum zy ( ) except for Borel summability.
Generally these approximations will not be accurate over the full range of the variable z. By an ‘educated’ choice
of zF( ), wemean building additional knowledge about the nature of zy ( ) into zF( ), whichmay lead to very
accurate approximations over the full range of the variable z evenwhen only a very limited number of
coefficients dk of the original asymptotic series are available. Typical examples of prior knowledge that can be
built into theΦ-Padé approximations are the large z behavior of zy ( ) or perhaps the large k behavior of the
coefficients dk.

2.1. The confluent hypergeometricΦ
Aprime candidate forΦ is the confluent hypergeometric functionU (see chapter 13 of [18]) or,more precisely,
the function

z z U a a b z, 1 , 1 , 14aF = + --( ) ( ) ( )

for which the coefficients fk in equation (2) are

f
a b

k
, 15k

k k=
( ) ( )

!
( )

where the Pochhammer symbol c k( ) is defined by c c k ck = G + G( ) ( ) ( ). Note that this zF( ) is symmetric in a
and b, which ismore obvious from equation (15) than from equation (14). From a theoretical point of view the
confluent hypergeometricU is a natural choice for at least two reasons. (i) the Borel–Padémethod is the special
case a b 1= = , since

z U z z E z1, 1, 1 e 1 , 16z1 1 1
1=- -( ) ( ) ( )

which is the E zEuler ( ) of equation (7). (ii) Just as the Borel transform is inverted by the Laplace transform, there is
a generalization (whichwe state without proof) that inverts the ‘confluent hypergeometric transform’ (see
equations (9) and (15)): if

z
d k

a b
z , 17

k

k

k k

k

0
åy =F
=

¥
ˆ ( ) !

( ) ( )
( )

then

z
a b

zs s U b a b s s
1

e 1 , 1, d . 18s a

0

1òy y=
G G

- - +
¥

F
- -( )

( ) ( )
ˆ ( ) ( ) ( )

(When b = 1,U a t0, , 1=( ) , and the result is the Borel–Leroy transformation [10].) From a practical point of
view, the confluent hypergeometric function(14) is also a very convenient choice, because as z  ¥,

z z
a b

a
z

b a

b
a b, integer , 19b aF ~

G -
G

+
G -
G

- ¹- -( ) ( )
( )

( )
( )

( ) ( )

z
z a

a
a b

log 2
, , 20a

0g y
~

- -
G

=- ( ) ( )
( )

( ) ( )
( )

where γ is Euler’s constant and a0y ( )( ) is the polygamma function. Since the approximant zn n, 1,yF - ( )[ ] depends
linearly onΦ (see equation (10)), an appropriate choice of a and b permits the large z behavior (if known) of zy ( )
to be built into theΦ-Padé approximants.We illustrate these general ideas with several examples and
generalizations of themethod.

3

J. Phys. Commun. 1 (2017) 025005 GÁlvarez andH J Silverstone



3. Examples

3.1. Zero-dimensional 4f field theory

As the simplest example, the confluent hypergeometric U , ,
g g

3

2

3 4 3

4

3

2

3

2
F = ( ) ( ) trivially sums the

perturbative series for the partition functionZ(g) of zero-dimensional 4f theory [9, 10], because

Z g x
1

2
e d , 21x gx2 42 4

òp
=

-¥

¥
- -( ) ( )!

g U g3 2 3 4, 3 2, 3 2 , 223 4=( ( )) ( ( )) ( )

is equal to theΦ of equation (14)with a 3 4= , b 1 4= and z g2 3= . In fact, the 0, 1[ ]approximant to the
asymptotic expansion ofZ(g),

Z g
k k

k

g2

3
, 23

k

k

0

3

4

1

4

3

4

1

4

å~
G + G +

G G
-

=

¥
⎜ ⎟⎛
⎝

⎞
⎠

( ) ( )
( ) ( )( )

!
( )

has z 3 21 = - , r 3 21 = , and is exactlyZ(g).

3.2. The Euler–Heisenberg effective Lagrangian
A second physically relevant example is the Euler–Heisenberg effective Lagrangian [20, 21]. For the spinor case
in a purelymagnetic background,

g s
s

s s

s
e coth

1

3

d
, 24s g

0 2
 ò= - -

¥
- ⎜ ⎟⎛

⎝
⎞
⎠( ) ( )

(see equations (1.18) and(1.19) in [21]), and has the asymptotic expansion,

g
B

k k k
g

2 4 2 3 2 2
2 , 25

k

k k

0

2 4 2 2 å~
+ + +=

¥
+ +( )

( )( )( )
( ) ( )

g g g
1

45

4

315

8

315
, 262 4 6~- + - + ( )

where B k2 4+ denote Bernouilli numbers. Standard Borel–Padé summation of equation (25)would involve Padé
approximants in g2 that lead to rational functions of s2, i.e., even functions of s, that have to approximate the
Borel transform, which is an odd function of s (essentially the non-exponential factor in the integrand of
equation (24)). This parity clash can be avoided by taking

z z U z2, 2, 1 , 272F = -( ) ( ) ( )

i.e., a=2, b=1, and f k 1k = +( )! rather than k!. The inverse confluent hypergeometric transform
equation (18) contains the explicit factor s, so that theΦ-transformwith a=2 and b=1 is in fact an even
function of s:

s s
s

s

s
coth

1

3

1
. 28a b, 2, 1 3

 = - -F = = ⎜ ⎟⎛
⎝

⎞
⎠ˆ ( ) ( )

For every n 1 , all the poles z j n, 1, 2, , ,j = ¼( ) of the n n1,-[ ]Padé approximants in s2 to sa b, 2, 1F = =
ˆ ( ) are

negative and simple,meaning that the poles in s are paired on the imaginary axis. The resulting approximants
have the form,

g
r

z
g z g z

1

2
i i , 29a b n n

j

n
j

j
j j, 2, 1; 1,

1

 å=
-

F - + F - -F = = -
=

( ) ( ( ) ( )) ( )[ ]

with the zF( ) given by equation (27). For example, thefirst Padé approximant to theΦ-transformed series is

g

g

g
g

45

21

2

1

45
1 45

4

315

1

3
, 30

2

2 21

2

2
2-

+
~ - - +⎜ ⎟⎛

⎝
⎞
⎠!

( )

with z1
21

2
= - , r

g
1 45

21

2

2

= - , and the correspondingΦ-Padé approximant is

g
g

g g
45

1

2
i 21 2 i 21 2 . 31a b, 2, 1; 0,1

2

 = - F + F -F = = ( ) ( ( ) ( )) ( )[ ]

If expanded as a power series in g, this simple approximation reproduces the first two nonvanishing terms of
equation (26), but at the same time it also captures the functional formof the large-g expansion: in fact

g g7 30 loga b, 2, 1; 0,1 ~ -F = = ( ) ( ) ( )[ ] , while the exact result is g g1 3 log ~ -( ) ( ) ( ) [21]. Note that the exact
expansion,

4

J. Phys. Commun. 1 (2017) 025005 GÁlvarez andH J Silverstone



s
s

s

s j j s
coth

1

3

1 2
, 32

j
3

1
2 2 2 2 2å p p

- - =
-

+=

¥
⎜ ⎟⎛
⎝

⎞
⎠ ( )

( )

can be viewed as the ‘ 1,¥ - ¥[ ]’Padé approximant in s2 for theΦ-transform, fromwhich the exact poles and
residues can be read off:

z j r
j

,
2

. 33j j
2 2

2 2
p

p
= - = - ( )

WithΦ given by equation (27), the resultingΦ-Padé infinite sum reproduces g( ):

g
j

g j g j2 i i

2
. 34a b

j
, 2, 1; 1,

1
4 4

 å p
p p

=
- F + F -

F = = ¥- ¥
=

¥

( ) ( ( )) ( ( )) ( )[ ]

We remark in passing that the coefficients j2 4 4p- ( ) give the rate of convergence of the approximants.

3.3.One-dimensional 4f field theory: the quartic anharmonic oscillator
Third, we consider one-dimensional 4f theory, i.e., the familiar x4-perturbed anharmonic oscillator, whose
Schrödinger equation is

x
x gx x E g x

1

2

d

d

1

2
. 35

2

2
2 4- + + Y = Y

⎛
⎝⎜

⎞
⎠⎟ ( ) ( ) ( ) ( )

Thefirst three coefficients of the ground state Rayleigh–Schrödinger perturbation series are

E g g g
1

2

3

4

21

8
. 362= + - + ( ) ( )

The coefficients E k( ) of this Borel-summable [3] series behave like

E k k1
2 3 1

2
, . 37k k

k
1

1 2

3 2

1
2

p
~ - G +  ¥+

+
⎜ ⎟⎛
⎝

⎞
⎠( ) ( )( )

More important is the large-g behavior ofE(g), which follows froma simple scaling argument,

E g g g, as , 381 3e~  ¥( ) ( )

where 0.667 986e = ¼ is the ground state energy of the purely quartic oscillator. If the g1 3 behavior is built
intoΦ, then even a two-parameter [0, 1] approximant gives an excellent fit toE(g) all theway from0 to¥. The
details are elementary enough to execute by hand. Because of the sign pattern, we sum the once-subtracted
series,

g
E g

g

1 2
, 39y =

-( ) ( ) ( )

whose large-g behavior is g 2 3- (thenmultiply by g and add 1/2 to report the results). Equation (19) shows that a
suitableΦwith this behavior can be obtained by taking a 2 3= and b a> in equation (14). If bwere then
chosen tofit the exact quartic ε, its value would be 0.997 7547¼.We take b=1 (Borel–Leroy–Padé but note
that a 2 3= is different from that implied by equation (37)). The 0, 1[ ]Padé approximant to the transformed
series, which needs only the two coefficients 3/4 and 21 8- from theE(g)-series and f 2 31 = from theΦ-
series, has z 4 211 = - and r 1 71 = . The [0, 1]Φ-Padé approximant is

E g g U
g

1

2

3

4

4

21

2

3
,

2

3
,

4

21
, 40, 0,1

2 3
1 3= +F ⎜ ⎟⎛

⎝
⎞
⎠

⎛
⎝⎜

⎞
⎠⎟( ) ( )[ ]

which, despite its simple origins, turns out to give remarkable agreement withE(g) for all g 0> , as seen in
figure 1. At¥,

E g g
3

4

4

21

1

3
, 41, 0,1

2 3
1 3~ GF ⎜ ⎟⎜ ⎟⎛

⎝
⎞
⎠

⎛
⎝

⎞
⎠( ) ( )[ ]

g g0.665 147 , as ; 421 3= ¼  ¥ ( )

the constant 0.665 147¼ is within 0.4%of the exact quartic ε. Higher-order n n1,-[ ]approximants generally
agree progressively better. It is clear fromfigure 1 inwhich the [0,1]Borel–Padé approximant is also plotted, how
relatively simple information used to choose the function generating thematch can dramatically affect the
quality and range of the approximant4.

4
All numerical calculations have been done in extended precision usingMathematica, version 11.1; the commands, PadeApproximant and

HypergeometricU, were particularly relevant.

5

J. Phys. Commun. 1 (2017) 025005 GÁlvarez andH J Silverstone



3.4. Implementation of the large-order behavior of the perturbation coefficients
As an example of the versatility of themethodwe showhow to incorporate in a simpleway the asymptotic
behavior of the coefficients dk into the functionΦ.We consider theβ-function for the 4f theory in d=3
dimensions [10], with coefficients

g g g g g

g g g O g

0
308

729
0.351 069 5977

0.376 526 8283 0.495 54751 0.749 689 , 43

2 3 4

5 6 7 8

b = - + - +

- + - +

˜ ( ˜) ˜ ˜ ˜ ˜

˜ ˜ ˜ ( ˜ ) ( )

and growth

k k k0.147 774 232 , . 44k
k 7 2b ~ - ¼  ¥˜ ( ) ! ( )

The [3, 4]Padé approximant for the Borel transformof gb̃( ˜) has a pole on the positive axis at g 17.34418=˜ and
consequently fails to be analytic in a strip containing the positive real axis, invalidating a possible [3, 4]Borel–
Padé approximant. Stirling’s formula shows that asymptotically the fk in equation (15) go like

f
k k

a b

a a b b

k
k1

1 6

2
, as , 45k

a b 2 2 2

~
G G

+
- + - +

 ¥
+ - ⎛

⎝⎜
⎞
⎠⎟

!
( ) ( )

( )

so that the growth of the coefficients kb̃ in equation (44) ismatchedwhen a b 11 2;+ = the k1 -term is then
minimumwhen a b 11 4= = .With this straightforward choice of a and b, andwith the corresponding [3, 4]
approximant to gb̃( ˜), we obtain a value for the nontrivial root of theβ-function of g 1.4192* =˜ , which is close
to the value 1.4105 of [10]. But we have no estimate of the accuracy of our calculation.

4.Φ-Padé approximants forGevrey-summable series

Nextwe adapt the newΦ-Padé approximantmethod to the cases of summable series whose coefficients dk grow
like mk( )!, where m 2, 3,= ¼, andwhich are variously known as generalized Borel summable [3],m-
summable orGevrey- m1 summable [2]. Them=2 case is useful for summing the x6-perturbed oscillator and
the Euler–Heisenberg series(25), andm=3 is useful for the x8-perturbed oscillator, etc.We regard these series
in zwith mk( )!growth to be series in z m1 with k!growth, but inwhich the coefficients of all the fractional
powers are 0. By averaging over them-th roots of unity, from a given (k!)- zF( ) (equation (2))we can constructm
appropriate ‘Gevrey- m1 ’ summed series zm1Fm ( )( ) , m0, 1, , 1m = ¼ - . zm1Fm ( )( ) has the asymptotic series,

z f z , 46m

k
mk

k1

0
åF ~ -m m
=

¥

+( ) ( ) ( )( )

and the explicit formula,

z
z

z

e

e
, 47m m j

m
m

j
m
j m m

m m
1

1

1
i 1

i 1

å w w
F =

F -
m

m p

p m
=

-

( )
( )

( )
( )( )

where em
m2 iw = p . The practical procedural consequence is that fk gets replaced by f mkm+ in equations (12) and

(13). The question, whichμ is appropriate, is similar towhich a and b are appropriate, and the answers depend
onwhich properties, e.g., large z, dk for large k, etc., aremost appropriate forψ.Moreover, the sameGevrey-1/m

Figure 1.ExactE(g) (black) and [0, 1] approximants for Borel–Padé (red) and a b2 3, 1= =( ) confluent hypergeometricΦ (blue).
The confluent hypergeometricΦ approximant agrees well with the exactE(g), because the g1 3 large-g behavior is carried by the gF( ).

6

J. Phys. Commun. 1 (2017) 025005 GÁlvarez andH J Silverstone



m1Fm
( ) can result from two different Gevrey-1Φʼs with differentμʼs, as illustrated in the next three equations and

following remark: if, for instance,

z k z , 48
k

k

0
åF ~ -
=

¥

( ) !( ) ( )

then

z k z2 , 49
k

k
0
1 2

0
åF ~ -
=

¥

( ) ( )!( ) ( )( )

z k z2 1 . 50
k

k
1
1 2

0
åF ~ + -
=

¥

( ) ( )!( ) ( )( )

The Euler–Heisenberg integral discussed above, particularly equation (29), is better understood as aGevrey-1/2
series summed by the 0m = version of the zF( ) given by equation (27), which is the same as the 1m = version
of z U z1, 1, 11- ( ) (equation (48)) given by equation (50).

4.1. The sextic anharmonic oscillator
A classicGevrey-1/2 series is the Rayleigh–Schrödinger perturbation series for the x6-perturbed anharmonic
oscillator (i.e, the Schrödinger equation (35)with gx4 replaced by gx6). Thefirst three coefficients of the ground-
state energy series are

E g g g
1

2

15

8

3495

64
. 512= + - + ( ) ( )

For large k, the coefficients E k( ) behave like

E k k1
32

2
1

2
, , 52k k

k
1

2

1

p
~ - G +  ¥+

+
⎜ ⎟⎜ ⎟

⎛
⎝

⎞
⎠

⎛
⎝

⎞
⎠( ) ( )( )

and for large g

E g g , 531 4e~( ) ( )

where εhere is the ground-state energy of the pure x6 oscillator. To build the g1 4 behavior into the
approximants, we take (for the once-subtracted series) z z U z3 2, 1, 13 2F = -( ) ( ). Although equation (19)
seems to imply that the large-z behavior would be z 1- rather than z 3 2- , the z 1- term is canceled in constructing

0
1 2F( ).When the approximant for the subtracted series ismultiplied by g, the remaining g1 2 3 2-( ) term gives

g1 4. The 0, 1[ ]Φ-Padé approximant, which like the x4 case can be done by hand, yields

E g g
g

1

2

15

8

30

233
. 54, 0,1 0

1 2= + FF

⎛
⎝⎜

⎞
⎠⎟( ) ( )[ ]

( )

This simple 0, 1[ ]approximation for the sextic oscillator, while superior to the 0, 1[ ]Borel–Padé approximant,
is not as dramatically accurate as the analogous approximation for the quartic oscillator given in equation (40),
but as n increases the accuracy of the n n1,-[ ]Φ-Padé approximant increasesmonotonically to the point that

Figure 2.ExactE(g) (black), Borel–Padé approximants (red), and a b3 2, 1= =( )-Padé approximants (blue) for the x6-perturbed
anharmonic oscillator. The g1 4 large-g behavior is carried by the a b3 2, 1= =( )-confluent-hypergeometric-function-based

g0
1 2F ( )( ) . The [0, 1] and [8, 9] approximants are shown. The largest relative error for the a b3 2, 1= =( ) [8, 9] approximant occurs

at g=100 and is less than 0.007, which is barely distinguishable from the exact E(g).

7

J. Phys. Commun. 1 (2017) 025005 GÁlvarez andH J Silverstone



infigure 2 it is difficult to distinguish between the exact and [8, 9]-approximant values for g0 100  . (The
maximum relative error at g = 100 is less than 0.007.)The error in the Borel–Padé approximants ismuch larger.

5. Summary

In summary, the conceptualization presented here emphasizesmatching the series to be summedwith a linear
combination of asymptotic series of known functions, cf equation (10). The known functions are scaled versions
of a single function zF( ), and the scaling and linear coefficients are calculated from the n n1,-[ ]Padé
approximants of the transformed series generated by zF( ). Thewhole idea stems from the realization that the
Borel–Padé approximant has exactly that structure, but where the zF( ) is the sumof Euler’s factorially divergent
power series, and from the thought that approximants would bemuchmore accurate if zF( )weremore
appropriate for the unknown sum zy ( ). Building the long-range behavior ofψ intoΦ is particularly successful.

Acknowledgments

Wewish to acknowledge the support of the SpanishMinisterio de Economía yCompetitividad under Project
No.FIS2015-63966-P and of theDepartment of Chemistry of the JohnsHopkinsUniversity.

References

[1] HardyGH1949Divergent Series (Oxford: Clarendon)
[2] Ramis J P 1993 Séries Divergentes et Théories Asymptotiques vol 121 (Marseille: SociétéMathématique de France)
[3] Graffi S, Grecchi V and SimonB 1970Phys. Lett.B 32 631
[4] SimonB 1970Ann. Phys. 58 76
[5] Herbst IW and SimonB 1978Phys. Lett. 78B 304
[6] ShrockR 2016Phys. Rev.D 94 125026
[7] Amore P 2007Phys. Rev.D 76 076001
[8] LeGuillou JC andZinn-Justin J 1977Phys. Rev. Lett. 39 95
[9] Zinn-Justin J 2002QuantumField Theory andCritical Phenomena 4th edn. (Oxford: Clarendon)
[10] Zinn-Justin J 2010Appl. Numer.Math. 60 1454
[11] BakerGA Jr, Nickel BG,GreenMS andMeironD I 1976Phys. Rev. Lett. 36 1351
[12] Franceschini V, Grecchi V and SilverstoneH J 1985Phys. Rev.A 32 1338
[13] ÁlvarezG,Martín-Mayor V andRuiz-Lorenzo J J 2000 J. Phys. A:Math. Gen. 33 841
[14] MeraH, Pedersen TG andNikolićBK2015Phys. Rev. Lett. 115 143001
[15] MeraH, Pedersen TG andNikolićBK2016Phys. Rev.B 94 165429
[16] Pedersen TG,MeraH andNikolićBK2016Phys. Rev.A 93 013409
[17] KleinertH and Schulte-Frohlinde V 2001Critical Properties of 4f -theories (Singapore:World Scientific)
[18] AbramowitzM and Stegun IA (ed) 1970Handbook ofMathematical Functions (NewYork:Dover)
[19] Euler L 1760Novi. Comm.Acad. Sci. Petrop. 5 205–37 (1754-55)
[20] HeisenbergWandEulerH 1936Z. Phys. 98 714
[21] DunneG2004Heisenberg-Euler effective Lagrangians: basics and extensions FromFields to Strings: Circumnavigating Theoretical

Physics (IanKoganMemorial Collection vol I) edMAShifman et al (Singapore:World Scientific) p 445

8

J. Phys. Commun. 1 (2017) 025005 GÁlvarez andH J Silverstone

https://doi.org/10.1016/0370-2693(70)90564-2
https://doi.org/10.1016/0003-4916(70)90240-X
https://doi.org/10.1016/0370-2693(78)90028-X
https://doi.org/10.1103/PhysRevD.94.125026
https://doi.org/10.1103/PhysRevD.76.076001
https://doi.org/10.1103/PhysRevLett.39.95
https://doi.org/10.1016/j.apnum.2010.04.002
https://doi.org/10.1103/PhysRevLett.36.1351
https://doi.org/10.1103/PhysRevA.32.1338
https://doi.org/10.1088/0305-4470/33/5/302
https://doi.org/10.1103/PhysRevLett.115.143001
https://doi.org/10.1103/PhysRevB.94.165429
https://doi.org/10.1103/PhysRevA.93.013409
https://doi.org/10.1007/BF01343663

	1. Introduction
	2.Φ-Padé approximants
	2.1. The confluent hypergeometric Φ

	3. Examples
	3.1. Zero-dimensional φ4 field theory
	3.2. The Euler–Heisenberg effective Lagrangian
	3.3. One-dimensional φ4 field theory: the quartic anharmonic oscillator
	3.4. Implementation of the large-order behavior of the perturbation coefficients

	4.Φ-Padé approximants for Gevrey-summable series
	4.1. The sextic anharmonic oscillator

	5. Summary
	Acknowledgments
	References