Asymptotic expansions for the psi function and the Euler-Mascheroni constant

Let r≠0 and s≠0 be two given real numbers. Chen [7] (2016) obtained recursive relation for determining the coefficients 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, the asymptotic expansion for the Euler-Mascheroni constant was derived. In this paper, we provide an explicit formula for these coefficients in terms of the cycle indicator polynomial of symmetric group which is an important tool in enumerative combinatorics. Also using this tool, we directly obtain an alternative form of the recursive relation for determining the coefficients aj(r,s). Furthermore we describe their asymptotic behavior for the special case r=2.