Resurgence in the Transition Region: The Incomplete Gamma Function

We study the resurgence properties of the coefficients C_n (τ) appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of C_n (τ) as n → +∞ depends on the parity of n. Both C_2n−1 (τ) and C_2n (τ) exhibit behaviours characterised by a leading term accompanied by an inverse factorial series, where the coefficients are once again C_2k−1 (τ) and C_2k (τ), 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, prior to this paper, there has been no investigation in the existing literature regarding the resurgence properties of asymptotic expansions in transition regions.