[Paper Review] A new type of factorial series expansions and applications

Ovidiu Costin, Rodica D. Costin|arXiv (Cornell University)|Aug 2, 2016

Mathematical functions and polynomials3 citations

TL;DR

This paper introduces dyadic factorial expansions—geometrically convergent asymptotic series that overcome the slow convergence and limited domain of classical factorial expansions. The method enables uniform, numerically efficient representations of special functions like Bessel, Airy, and Ei across the complex plane, including Stokes rays, and extends to Écalle resurgent functions and resolvent decompositions of self-adjoint operators.

ABSTRACT

We construct a new type of convergent asymptotic representations, dyadic factorial expansions. Their convergence is geometric and the region of convergence can include Stokes rays, and often extends down to 0^+. For special functions such as Bessel, Airy, Ei, Erfc, Gamma and others, this region is C without an arbitrarily chosen ray effectively providing uniform convergent asymptotic expansions for special functions. We prove that relatively general functions, Ecalle resurgent ones possess convergent dyadic factorial expansions. We show that dyadic expansions are numerically efficient representations. The expansions translate into representations of the resolvent of self-adjoint operators in series in terms of the associated unitary evolution operator evaluated at some prescribed points (alternatively, in terms of the generated semigroup for positive operators).

Motivation & Objective

• Address the limitations of classical factorial expansions, which suffer from slow (power-like) convergence and restricted domains that exclude Stokes rays.
• Develop a new class of convergent factorial expansions—dyadic factorial expansions—that achieve geometric convergence and extend to regions including the origin and Stokes rays.
• Provide a unified framework for asymptotic expansions of special functions (e.g., Ei, Airy, Bessel, Gamma) with uniform convergence in the complex plane, excluding only an arbitrarily chosen ray.
• Establish a connection between dyadic expansions and resolvent decompositions of self-adjoint and positive operators via unitary evolution or semigroups.
• Generalize the theory to Écalle resurgent functions, showing that their divergent series can be resummed using these new expansions.

Proposed method

• Construct a dyadic decomposition of the Cauchy kernel using a recursive partitioning of the complex plane based on dyadic intervals.
• Apply this decomposition to the integral representation of the exponential integral Ei⁺(x), leading to a series in terms of scaled Pochhammer symbols (x)_k.
• Use Laplace transforms and integration by parts to derive factorial series expansions with remainders that decay geometrically.
• Leverage the properties of the polylogarithm function and its derivatives, expressed via Stirling numbers of the first kind, to generate explicit coefficients.
• Establish a general theory for dyadic expansions of Écalle resurgent functions by decomposing them into resurgent elements and analyzing their Borel transforms.
• Derive resolvent decompositions for self-adjoint operators by expressing the resolvent as a series in the unitary evolution operator at dyadic time points.

Experimental results

Research questions

• RQ1 Can dyadic factorial expansions be constructed such that their convergence is geometric and their domain includes Stokes rays, even extending down to 0⁺?
• RQ2 How can the classical limitations of slow convergence and restricted convergence domains in factorial expansions be overcome using a dyadic decomposition of the Cauchy kernel?
• RQ3 To what extent can dyadic expansions be applied uniformly across the complex plane for special functions like Ei, Airy, and Bessel functions?
• RQ4 What is the connection between dyadic factorial expansions and the resolvent of self-adjoint or positive operators, and how can this be expressed in terms of unitary evolution or semigroups?
• RQ5 Can the theory of Écalle resurgent functions be extended to include geometrically convergent dyadic expansions, and how does this relate to Borel summability?

Key findings

• Dyadic factorial expansions achieve geometric convergence, significantly outperforming classical factorial series that converge only power-like.
• The region of convergence for the dyadic expansions of Ei⁺(x) includes the entire complex plane minus the negative real axis, effectively covering the Stokes ray at ℝ⁺.
• For special functions such as Bessel, Airy, Ei, Erfc, and Gamma, the expansions converge uniformly on ℂ minus an arbitrarily chosen ray, enabling robust asymptotic approximations.
• The method successfully produces a dyadic expansion for the polylogarithm function Li₁/₂(z), with coefficients derived from Stirling numbers and higher derivatives of the polylogarithm.
• The remainder terms in the expansions decay geometrically, as shown by the bounds on R_n and R_nk, which are O(1/(y)_{n-1}) and O(1/(2^k y)_{n-1}) respectively, ensuring rapid convergence.
• The theory extends to Écalle resurgent functions, proving that their divergent series can be resummed via dyadic factorial expansions, and that such expansions are compatible with Borel summation.