engleski

New asymptotic expansions and approximation formulas for the factorial function

Well-known Stirling approximation for the factorial function is one of the most beautiful formulas in mathematics. This is in fact a shortening of the asymptotic expansion for the gamma function which was also studied in other similar forms by Laplace, De Moivre, Ramanujan, Wehmeier and recently by Karatsuba, Gosper, Batir, Mortici, Nemes and others. But all this formulas have been considered separately and connection between them was not clear. Moreover, the compution of each term in this formulas was a tedious job without any attempt to find general procedure to calculate coefficients in this type of asymptotic expansions. In this talk we present general expansion for the gamma function introducing parameter m. Using properties of asymptotic power series, we proved asymptotic expansion for the factorial function where coefficients (Pn) satisfy simple recursive algorithm. This allows easy calcuation of the coefficients in all of the previously mentioned expansions and leads to various other generalizations and improvements of the approximation formulas for the gamma function and related classical functions.

gamma function ; factorial function ; asymptotic expansions

nije evidentirano