Unified approaches to the approximations of the gamma function

In this paper, by the exponential complete Bell polynomials, we establish two general approximations for the gamma function, where the coefficients in the series of the approximations can be determined by recurrences. These two general approximations include as special cases some well-known results such as those under the names De Moivre, Gosper, Gosper–Smith, Laplace, Luschny, Ramanujan and Wehmeier, and some recently published results such as those due to Batir, Chen, Lu, Mortici, Nemes, Paris, et al. By these two general approximations, we give unified approaches to dealing with many known approximations of the gamma function and to establishing new ones.