Divergent Power Series Research Articles

Peacock patterns and resurgence in complex Chern–Simons theory

The partition function of complex Chern–Simons theory on a 3-manifold with torus boundary reduces to a finite-dimensional state-integral which is a holomorphic function of a complexified Planck’s constant tau in the complex cut plane and an entire function of a complex parameter u. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear q-difference equation. We further conjecture that a distinguished entry of the Stokes automorphism matrix is the 3D-index of Dimofte–Gaiotto–Gukov. We provide proofs of our statements regarding the q-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic textbf{4}_1 and textbf{5}_{2} knots.

Asymptotically consistent analytical solutions for the non-Newtonian Sakiadis boundary layer

The Sakiadis boundary layer induced by a moving wall in a semi-infinite fluid domain is a fundamental laminar flow field relevant to high speed coating processes. This work provides an analytical solution to the boundary-layer problem for Ostwald–de Waele power law fluids via a power series expansion and extends the approach taken for Newtonian fluids [Naghshineh et al. “On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs,” IMA J. of Appl. Math. 88, 43 (2023)] in which variable substitutions (which naturally determine the gauge function in the power series) are chosen to be consistent with the large distance behavior away from the wall. Contrary to prior literature, the asymptotic behavior dictates that a solution only exists in the range of power law exponents, α, lying in the range of 0.5<α≤1. An analytical solution is obtained in the range of approximately 0.74≤α<1, using a convergent power series with an asymptotically motivated gauge function. For power laws corresponding to 0.5<α<0.74, the gauge function becomes ill-defined over the full domain, and an approximate analytical solution is obtained using the method of asymptotic approximants [Barlow et al. “On the summation of divergent, truncated, and underspecified power series via asymptotic approximants,” Q. J. Mech. Appl. Math. 70, 21–48 (2017)]. The approximant requires knowledge of two physical constants, which we compute a priori using a numerical shooting method on a finite domain. The utility of the power series solution is that it can be solved on the entire semi-infinite domain and—in contrast to a numerical solution—does not require a finite domain length approximation and subsequent domain length refinement.

Dyson–Schwinger equations in minimal subtraction

We compare the solutions of one-scale Dyson–Schwinger equations (DSEs) in the minimal subtraction (MS) scheme to the solutions in kinematic momentum subtraction (MOM) renormalization schemes. We establish that the MS-solution can be interpreted as a MOM-solution, but with a shifted renormalization point, where the shift itself is a function of the coupling. We derive relations between this shift and various renormalization group functions and counterterms in perturbation theory. As concrete examples, we examine three different one-scale Dyson–Schwinger equations: one based on the 1-loop multiedge graph in D=4 dimensions, one for D=6 dimensions, and one for mathematical toy model. For each of the integral kernels, we examine both the linear and nine different non-linear Dyson–Schwinger equations. For the linear cases, we empirically find exact functional forms of the shift between MOM and MS renormalization points. For the non-linear DSEs, the results for the shift suggest a factorially divergent power series. We determine the leading asymptotic growth parameters and find them in agreement with the ones of the anomalous dimension. Finally, we present a tentative exact solution to one of the non-linear DSEs of the toy model.

The Resurgent Structure of Quantum Knot Invariants

The asymptotic expansion of quantum knot invariants in complex Chern–Simons theory gives rise to factorially divergent formal power series. We conjecture that these series are resurgent functions whose Stokes automorphism is given by a pair of matrices of q-series with integer coefficients, which are determined explicitly by the fundamental solutions of a pair of linear q-difference equations. We further conjecture that for a hyperbolic knot, a distinguished entry of those matrices equals to the Dimofte–Gaiotto–Gukov 3D-index, and thus is given by a counting of BPS states. We illustrate our conjectures explicitly by matching theoretically and numerically computed integers for the cases of the 4_1 and the 5_2 knots.

Analytic solution of the SEIR epidemic model via asymptotic approximant

An analytic solution is obtained to the SEIR Epidemic Model. The solution is created by constructing a single second-order nonlinear differential equation in lnS and analytically continuing its divergent power series solution such that it matches the correct long-time exponential damping of the epidemic model. This is achieved through an asymptotic approximant (Barlow et al., 2017) in the form of a modified symmetric Padé approximant that incorporates this damping. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.

Accurate closed-form solution of the SIR epidemic model

An accurate closed-form solution is obtained to the SIR Epidemic Model through the use of Asymptotic Approximants (Barlow et al., 2017). The solution is created by analytically continuing the divergent power series solution such that it matches the long-time asymptotic behavior of the epidemic model. The utility of the analytical form is demonstrated through its application to the COVID-19 pandemic.

On the summability of divergent power series satisfying singular PDEs

The aim of this note is to apply the Borel–Laplace summation method studied by H. Chen, Z. Luo and C. Zhang (Summability of formal solutions of singular PDEs by means of two-dimensional Borel–Laplace method, preprint) to the divergent power series solutions to two families of nonlinear PDEs. The first one contains particularly a two-dimensional version of the so-called Euler equation (ODE), while the second is called totally characteristic type PDE by H. Chen and H. Tahara (On the holomorphic solution of non-linear totally characteristic equations, Math. Nachr. 219 (2000) 85–96).

Application of the Cauchy integral formula as a tool of analytic continuation for the resummation of divergent perturbation series

Previous attempts to the resummation of divergent power series by means of analytic continuation are improved applying the Cauchy integral formula for complex functions. The idea is tested on divergent Møller-Plesset perturbation expansions of the electron correlation energy. In particular, the potential curve of the LiH molecule is computed from single reference MPn results which are divergent for bond distances larger than 3.6 Å. Preliminary results for the Hartree-Fock molecule are also tabulated.

Factorial Transformation for Some Classical Combinatorial Sequences

Factorial transformation known from Euler’s time is a very powerful tool for summation of divergent power series. We use factorial series for summation of ordinary power generating functions for some classical combinatorial sequences. These sequences increase very rapidly, so OGFs for them diverge and mostly unknown in a closed form. We demonstrate that factorial series for them are summable and expressed in known functions. We consider among others Stirling, Bernoulli, Bell, Euler and Tangent numbers. We compare factorial transformation with other summation techniques such as Pade approximations, transformation to continued fractions, and Borel integral summation. This allowed us to derive some new identities for GFs and express their integral representations in a closed form.

Pluripolar Hulls and Convergence Sets

The pluripolar hull of a pluripolar set E in ℙn is the intersection of all complete pluripolar sets in ℙn that contain E. We prove that the pluripolar hull of each compact pluripolar set in ℙn is Fσ. The convergence set of a divergent formal power series f(z0, … , zn) is the set of all “directions” ξ ∈ ℙn along which f is convergent. We prove that the union of the pluripolar hulls of a countable collection of compact pluripolar sets in ℙn is the convergence set of some divergent series f. The convergence sets on Γ := {[1 : z : ψ (z)] : z ∈ ℂ} ⊂ ℂ2 ⊂ ℙ2, where ψ is a transcendental entire holomorphic function, are also studied and we obtain that a subset on Γ is a convergence set in ℙ2 if and only if it is a countable union of compact projectively convex sets, and hence the union of a countable collection of convergence sets on Γ is a convergence set.

Quasimodularity and large genus limits of Siegel-Veech constants

Quasimodular forms were first studied systematically in the context of counting torus coverings. Here we show that a weighted version of these coverings with Siegel-Veech weights also provides quasimodular forms. We apply this to prove conjectures of Eskin and Zorich on the large genus limits of Masur-Veech volumes and of Siegel-Veech constants. In Part I we connect the geometric definition of Siegel-Veech constants both with a combinatorial counting problem and with intersection numbers on Hurwitz spaces. We also introduce certain modified Siegel-Veech weights whose generating functions will later be shown to be quasimodular. Parts II and III are devoted to the study of the (quasi) modular properties of the generating functions arising from weighted counting of torus coverings. These two parts contain little geometry and can be read independently of the rest of the paper. The starting point is the theorem of Bloch and Okounkov saying that certain weighted averages, called q q -brackets, of shifted symmetric functions on partitions are quasimodular forms. In Part II we give an expression for the growth polynomials (a certain polynomial invariant of quasimodular forms) of these q q -brackets in terms of Gaussian integrals, and we use this to obtain a closed formula for the generating series of cumulants that is the basis for studying large genus asymptotics. In Part III we show that the even hook-length moments of partitions are shifted symmetric polynomials, and we prove a surprising formula for the q q -bracket of the product of such a hook-length moment with an arbitrary shifted symmetric polynomial as a linear combination of derivatives of Eisenstein series. This formula gives a quasimodularity statement also for the ( − 2 ) (-2) -nd hook-length moments by an appropriate extrapolation, and this in turn implies the quasimodularity of the Siegel-Veech weighted counting functions. Finally, in Part IV these results are used to give explicit generating functions for the volumes and Siegel-Veech constants in the case of the principal stratum of abelian differentials. The generating functions have an amusing form in terms of the inversion of a power series (with multiples of Bernoulli numbers as coefficients) that gives the asymptotic expansion of a Hurwitz zeta function. To apply these exact formulas to the Eskin-Zorich conjectures on large genus asymptotics (both for the volume and the Siegel-Veech constant) we provide in a separate appendix a general framework for computing the asymptotics of rapidly divergent power series.

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 Φ. Both the scaling and linear coefficients are calculated from Padé approximants of a series transformed from the original series by Φ. We discuss in particular the case that Φ is (essentially) a confluent hypergeometric function, which includes as special cases the standard Borel–Padé and Borel–Leroy–Padé 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–Schrödinger perturbation theory illustrate the method.

On the Summation of Divergent, Truncated, and Underspecified Power Series via Asymptotic Approximants

A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recursively on this coefficient are also unknown. The constructed approximants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approximants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.

Factorial transformation for some classical combinatorial sequences

Factorial transformation known from Euler’s time is a very powerful tool for summation of divergent power series. We use factorial series for summation of ordinary power generating functions for some classical combinatorial sequences. These sequences increase very rapidly, so OGFs for them diverge and mostly unknown in a closed form. We demonstrate that factorial series for them are summable and expressed in known functions. We consider among others Stirling, Bernoulli, Bell, Euler and Tangent numbers. We compare factorial transformation with other summation techniques such as Pade approximations, transformation to continued fractions, and Borel integral summation. This allowed us to derive some new identities for GFs and express their integral representations in a closed form.

Euler's divergent series and an elementary model in statistical physics

We discuss the multiple integral of a multivariate exponential taken with respect either to the Lebesgue measure or to the discrete uniform Bernoulli measure. In the first case the integral is linked to Euler's everywhere divergent power series and its generalizations, while in the second case the integral is linked to a one-dimensional model of spin systems as encountered in physics.

On convergence sets of formal power series

The convergence set of a divergent formal power series f ( x 0 , ⋯ , x n ) is the set of all “directions” ξ ∈ P n along which f is absolutely convergent. We prove that every countable union of closed complete pluripolar sets in P n is the convergence set of some divergent series f. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional analogs of the results of Sathaye (J Reine Angew Math 283:86–98, 1976 ), Lelong (Proc Am Math Soc 2:11–19, 1951 ), Levenberg and Molzon (Math Z 197:411–420, 1988 ), and of Ribon (Ann Scuola Norm Sup Pisa Cl Sci (5) 3:657–680 2004 ) are obtained.

On the summability of divergent power series solutions for certain first-order linear PDEs

This article is concerned with the study of the Borel summability of divergent power series solutions for certain singular first-order linear partial differential equations of nilpotent type. Our main purpose is to obtain conditions which coefficients of equations should satisfy in order to ensure the Borel summability of divergent solutions. We will see that there is a close affinity between the Borel summability of divergent solutions and global analytic continuation properties for coefficients of equations.

Confluence of meromorphic solutions of q-difference equations

Dans cet article, nous considérons un q-analogue du processus de sommation de Borel-Laplace, avec q>1 paramètre réel. En particulier, nous prouvons que la sommation de Borel-Laplace d’une série formelle solution d’une équation différentielle linéaire peut être approchée, dans un secteur convenable, par une solution méromorphe d’une certaine famille d’équations aux q-différences linéaire. Nous faisons les calculs pour les séries hypergéométriques. En s’inspirant de Sauloy, nous prouvons comment une base de solutions d’une equation différentielle linéaire peut être approchée, sur un secteur convenable, par une base de solutions d’une famille correspondante d’équations aux q-différences. Cela nous mène à l’approximation des matrices de Stokes et de monodromies de l’équation différentielle, par des matrices dont les entrées sont invariantes par multiplication par q.

An example of Newton's method for an equation in Gevrey series

In the context of complex WKB analysis, we discuss a one-dimensional Schrödinger equation−h2∂x2f(x,h)+[Q(x)+hQ1(x,h)]f(x,h)=0,h→0, where Q(x), Q1(x,h) are analytic near the origin x=0, Q(0)=0, and Q1(x,h) is a factorially divergent power series in h. We show that there is a change of independent variable y=y(x,h), analytic near x=0 and factorially divergent with respect to h, that transforms the above Schrödinger equation to a canonical form. The proof goes by reduction to a mildly nonlinear equation on y(x,h) and by solving it using an appropriately modified Newton's method of tangents.

On classes of everywhere divergent power series

We prove the everywhere divergence of series $$ \sum_{n=0}^\infty a_n e^{i\rho_n}e^{inx}, \quad\text{and}\quad \sum_{n=0}^\infty {(-1)}^{[\rho_n]}a_n \cos nx, $$ for sequences an and ρn satisfying some extremal conditions. These results generalize some well known examples of everywhere divergent power and trigonometric series.