Sharp inequalities and asymptotic expansions for the gamma function

In this paper, we give a recurrence relation for determining the coefficients ajaj such thatΓ(x+1)∼2πx(xe)x(1+112x+∑j=0∞ajx−j),x→∞. We provide a pair of recurrence relations for determining the constants αℓαℓ and βℓβℓ such thatΓ(x+1)∼2πx(xe)x(1+∑ℓ=1∞αℓ(x+βℓ)2ℓ−1),x→∞. Furthermore, we determine the best possible constants a and b such that the inequality2πn(ne)n(1+112n−a)≤n!<2πn(ne)n(1+112n−b) holds for all integers n≥1n≥1.