New approximations of the gamma function in terms of the digamma function

The goal of this paper is to prove the following asymptotic formula Γ(x)≈2πe−b(x+b)xexp(−x−12ψ(x+c)) as x∈N,x→∞, where Γ is the Euler Gamma function and ψψ is the digamma function, namely, the logarithmic derivative of Γ. Moreover, optimal values of parameters b,cb,c are calculated in such a way that this asymptotic convergence is the best possible.