Padé approximant related to asymptotics for the gamma function

摘要

Based on the Padé approximation method, we determine the coefficients aj and bj (1 ≤ j ≤ k) such that Γ(x+1)2πx(x/e)x=xk+a1xk−1+⋯+akxk+b1xk−1+⋯+bk+O(1x2k+1),x→∞, where k ≥ 1 is any given integer. Based on the obtained result, we establish new bounds for the gamma function.