Polynomials and asymptotic constants in a resurgent problem from ’t Hooft

In a recent study of the quantum theory of harmonic oscillators, Gerard ’t Hooft proposed the following problem: given G(z)=∑n=1∞nzn for |z|<1, find its analytic continuation for |z|≥1, excluding a branch-cut z∈[1,∞). A solution is provided by the bilateral convergent sum G(z)=12π∑n=-∞∞(2πin-log(z))-3/2. On the negative real axis, G(-eu) has a sign-constant asymptotic expansion in 1/u2, for large positive u. Optimal truncation leaves exponentially suppressed terms in an asymptotic expansion e-u∑k=0∞Pk(x)/uk, with P0(x)=x-23 and Pk(x) of degree 2k+1 evaluated at x=u/2-⌊u/2⌋. At large k, these polynomials become excellent approximations to sinusoids. The amplitude of Pk(x) increases factorially with k and its phase increases linearly, with Pk(x)∼sin((2k+1)C-2πx)R2k+1Γ(k+12)/2π, where C≈1.0688539158679530121571 and R≈0.5181839789815558726739 are asymptotic constants satisfying Rexp(iC)=-1/(2+πi).