A new method to sum divergent power series: educated match

We present a method to sum Borel- and Gevrey-summable asymptotic series by matching the series to be summed with a linear combination of asymptotic series of known functions that themselves are scaled versions of a single, appropriate, but otherwise unrestricted, function $\Phi$. Both the scaling and linear coefficients are calculated from Pad\'e approximants of a series transformed from the original series by $\Phi$. We discuss in particular the case that $\Phi$ is (essentially) a confluent hypergeometric function, which includes as special cases the standard Borel-Pad\'e and Borel-Leroy-Pad\'e methods. A particular advantage is the mechanism to build knowledge about the summed function into the approximants, extending their accuracy and range even when only a few coefficients are available. Several examples from field theory and Rayleigh-Schr\"odinger perturbation theory illustrate the method.


Introduction
Summation of divergent asymptotic expansions has led to a vast literature from both mathematical and physical points of view. The mathematical goal is often to assign a standard sum to a series whose coefficients satisfy certain growth conditions and whose sum satisfies certain conditions at infinity [1,2]. The physical literature focuses on a wide range of specialized, computational methods. Especially since the work carried out in the 1970s on the coupling constant analyticity of anharmonic oscillators [3][4][5], two summation methods have become dominant: Padé approximants and Borel summation. Both have been found useful in fields as diverse as quantum mechanics, statistical mechanics, quantum field 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 4 lF theory [6]), and at times with 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 two ways: conformal mappings [8][9][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][12][13]. Most recently, Mera et al [14,15] and Pedersen et al [16] have developed a method that uses hypergeometric functions to sum perturbation theory series using only a few terms.
Initially motivated in part by the papers of Mera et al, we present here a new method to build concise, explicit, analytic approximants to the Borel or Gevrey sum of an asymptotic power series. These approximants match the series to be summed with a linear combination of asymptotic series of known functions. The known functions are scaled versions of a single function Φ, and both the scaling and linear coefficients are readily calculated from Padé approximants of a transformed series determined by the original series and by Φ. If Φ is taken to be (essentially) a confluent hypergeometric function, the new method includes as special cases the standard Borel-Padé and Borel-Leroy-Padé summation methods. Even more important, prior additional (i.e., educated) knowledge about the summed function can be built into the approximants via the function Φ, 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 're-expanding the asymptotic expansion in a complete set of basis functions', and which is mathematically equivalent to conformal mapping techniques [17]. Our method, in contrast, is essentially linked to the theory of Padé approximants.

Φ-Padé approximants
Our goal is to approximate the Borel sum z y ( ) of a divergent power series, using any appropriate known function z F( ) with its own Borel-summable series, The method is at the same time hidden in, and a generalization of, the Borel-Padé summation method [13], which we briefly review. Let us denote by P z Q z n n 1 ]Padé approximant of the Borel transform of the series (1), and let us assume that Q n (z) has only simple zeros. Partial fraction expansion, and where E z 1 1 ( ) 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 z Euler ( ) in equation (7) is precisely the Borel sum of the factorially divergent Euler series [19] obtained by setting f k k = ! in equation (2), and (ii) that the asymptotic expansion of z n n B, 1, is identical to that of z y ( ) through order z n 2 1 -, i.e., that As a generalization of equation (8), the approximants z n n , 1, , which depend on n 2 parameters r z j n , , 1, , and therefore the r j and z j solve the n 2 equations In practice, these parameters are most easily calculated from the partial fraction expansion of the n n 1, In other words, the z j are the poles, for simplicity assumed to be simple, and the r j the residues, of the n n 1, The Borel-Padé approximant uses no information about the sum z y ( ) except for Borel summability.
Generally these approximations will not be accurate over the full range of the variable z. 2.1. The confluent hypergeometric Φ A prime candidate for Φ is the confluent hypergeometric function U (see chapter 13 of [18]) or, more precisely, the function for which the coefficients f k in equation (2) are Note that this z F( ) is symmetric in a and b, which is more obvious from equation (15) than from equation (14). From a theoretical point of view the confluent hypergeometric U is a natural choice for at least two reasons. (i) the Borel-Padé method is the special (7). (ii) Just as the Borel transform is inverted by the Laplace transform, there is a generalization (which we state without proof) that inverts the 'confluent hypergeometric transform' (see equations (9) and (15) , 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  ¥, where γ is Euler's constant and a , and is exactly Z(g).  (25) would involve Padé approximants in g 2 that lead to rational functions of s 2 , 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

The Euler-Heisenberg effective Lagrangian
, while the exact result is g g 1 3 log ~-( ) ( ) ( ) [21]. Note that the exact expansion, ] ' Padé approximant in s 2 for the Φ-transform, from which the exact poles and residues can be read off: With Φ given by equation (27), the resulting Φ-Padé infinite sum reproduces g ( ): We remark in passing that the coefficients j 2 4 4 p -( ) give the rate of convergence of the approximants.

3.3.
One-dimensional 4 f field theory: the quartic anharmonic oscillator Third, we consider one-dimensional 4 f theory, i.e., the familiar x 4 -perturbed anharmonic oscillator, whose Schrödinger equation is The first three coefficients of the ground state Rayleigh-Schrödinger perturbation series are The coefficients E k ( ) of this Borel-summable [3] series behave like More important is the large-g behavior of E(g), which follows from a simple scaling argument, where 0.667 986 e = ¼is the ground state energy of the purely quartic oscillator. If the g 1 3 behavior is built into Φ, then even a two-parameter [0, 1] approximant gives an excellent fit to E(g) all the way from 0 to ¥. The details are elementary enough to execute by hand. Because of the sign pattern, we sum the once-subtracted series, whose large-g behavior is g 2 3 -(then multiply by g and add 1/2 to report the results). Equation (19) shows that a suitable Φ with this behavior can be obtained by taking a 2 3 = and b a > in equation (14). If b were then chosen to fit 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 the E(g)-series and f 2 3 1 = from the Φseries, has z 4 21 1 =and r 1 7 which, despite its simple origins, turns out to give remarkable agreement with E(g) for all g 0 > , as seen in figure 1. At ¥,

Implementation of the large-order behavior of the perturbation coefficients
As an example of the versatility of the method we show how to incorporate in a simple way the asymptotic behavior of the coefficients d k into the function Φ. We consider the β-function for the 4 f theory in d=3 dimensions [10], with coefficients , which is close to the value 1.4105 of [10]. But we have no estimate of the accuracy of our calculation.

Φ-Padé approximants for Gevrey-summable series
Next we adapt the new Φ-Padé approximant method to the cases of summable series whose coefficients d k grow like mk ( )!, where m 2, 3, = ¼ , and which are variously known as generalized Borel summable [3], msummable or Gevreym 1 summable [2]. The m=2 case is useful for summing the x 6 -perturbed oscillator and the Euler-Heisenberg series(25), and m=3 is useful for the x 8 -perturbed oscillator, etc. We regard these series in z with mk ( )! growth to be series in z m 1 with k! growth, but in which the coefficients of all the fractional powers are 0. By averaging over the m-th roots of unity, from a given (k!)z F( ) (equation (2)) we can construct m has the asymptotic series,  (12) and (13). The question, which μ is appropriate, is similar to which a and b are appropriate, and the answers depend on which properties, e.g., large z, d k for large k, etc., are most appropriate for ψ. Moreover, the same Gevrey-1/m ) confluent hypergeometric Φ (blue). The confluent hypergeometric Φ approximant agrees well with the exact E(g), because the g 1 3 large-g behavior is carried by the g F( ).
, the z 1 term is canceled in constructing  . The [0, 1] and [8,9] approximants are shown. The largest relative error for the a b 3 2, 1 = = ( ) [8,9] approximant occurs at g=100 and is less than 0.007, which is barely distinguishable from the exact E(g).
in figure 2 it is difficult to distinguish between the exact and [8,9]-approximant values for g 0 100   . (The maximum relative error at g = 100 is less than 0.007.) The error in the Borel-Padé approximants is much larger.

Summary
In summary, the conceptualization presented here emphasizes matching the series to be summed with a linear combination of asymptotic series of known functions, cf equation (10). The known functions are scaled versions of a single function z F( ), and the scaling and linear coefficients are calculated from the n n 1, -[ ]Padé approximants of the transformed series generated by z F( ). The whole idea stems from the realization that the Borel-Padé approximant has exactly that structure, but where the z F( ) is the sum of Euler's factorially divergent power series, and from the thought that approximants would be much more accurate if z F( ) were more appropriate for the unknown sum z y ( ). Building the long-range behavior of ψ into Φ is particularly successful.