Inequalities and asymptotic expansions for the psi function and the Euler–Mascheroni constant

Let r≠0r≠0 and s≠0s≠0 be two given real numbers. We give a recursive relation for determining the coefficients aj(r,s)aj(r,s) such thatψ(x+1)∼ln⁡x+(1−1r)1x+1sln⁡(1+∑j=1∞aj(r,s)xj),x→∞, where ψ denotes the psi function. As a consequence, we obtain asymptotic expansion for the Euler–Mascheroni constant. Furthermore, we present new inequalities for the psi function. As a consequence, we obtain the higher order estimate for the Euler–Mascheroni constant.