Resurgence in the Transition Region: The Incomplete Gamma Function - Nemes, Gergő (Symmetry, Integrability and Geometry: Methods and Applications, 2024)
We study the resurgence properties of the coefficients ₙ() appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of ₙ() as → +∞ depends on the parity of . Both ₂ₙ₋₁() and ₂ₙ(τ) exhibit behaviours characterised by a leading term accompanied by an inverse factorial series, where the coefficients are once again ₂ₖ₋₁() and ₂ₖ(), respectively. Our derivation employs elementary tools and relies on the known resurgence properties of the asymptotic expansion of the gamma function and the uniform asymptotic expansion of the incomplete gamma function. To the best of our knowledge, before this paper, there has been no investigation in the existing literature regarding the resurgence properties of asymptotic expansions in transition regions.