| Popis: |
Mortici (2015) [31] proposed a new formula for approximating the gamma function and the convergence of the corresponding asymptotic series is very fast in comparison with other classical or recently discovered asymptotic series. In this paper, by the Lagrange–Burmann formula we give an explicit formula for determining the coefficients a k ( k = 1 , 2 , … ) in Mortici's formula such that Γ ( x + 1 ) 2 π x ( x e ) x ∼ exp { ∑ k = 1 ∞ a k ( x 12 x 2 + 2 5 ) k } , x → ∞ . Moreover, by the cycle indicator polynomial of symmetric group, we give an explicit expression for the coefficients b k ( k = 0 , 1 , … ) of the following expansion: Γ ( x + 1 ) 2 π x ( x e ) x ∼ ( ∑ k = 0 ∞ b k ( x 12 x 2 + 2 5 ) k ) 1 / r , x → ∞ . A recursive formula for calculating the coefficients b k ( k = 0 , 1 , … ) is also given. |
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