Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4593191 | Journal of Number Theory | 2016 | 10 Pages |
Abstract
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–Bürmann formula we give an explicit formula for determining the coefficients akak(k=1,2,…)(k=1,2,…) in Mortici's formula such thatΓ(x+1)2πx(xe)x∼exp{∑k=1∞ak(x12x2+25)k},x→∞. Moreover, by the cycle indicator polynomial of symmetric group, we give an explicit expression for the coefficients bkbk(k=0,1,…)(k=0,1,…) of the following expansion:Γ(x+1)2πx(xe)x∼(∑k=0∞bk(x12x2+25)k)1/r,x→∞. A recursive formula for calculating the coefficients bkbk(k=0,1,…)(k=0,1,…) is also given.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Aimin Xu, Yongcai Hu, Peipei Tang,