Two asymptotic expansions for gamma function developed by Windschitl's formula

2017/12/21 by Yang, Zhen-Hang, Tian, Jing-Feng
#33B15 #41A10 #41A20 (Secondary) #41A60 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1712.08039

Abstract

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for n∈ ℕ with n≥ 4, we have Γ( x+1) =√(2πx)( \tfracxe) ^x( x\sinh \tfrac1x) ^x/2exp ( ∑_k=3^n-1\tfrac( 2k( 2k-2) !-2^2k-1) B_2k2k( 2k) !x^2k-1 +R_n( x) ) with | R_n( x) | ≤ \frac| B_2n| 2n( 2n-1) \frac1x^2n-1 for all x>0, where B_2n is the Bernoulli number. Moreover, we present some approximation formulas for gamma function related to Windschitl's approximation one, which have higher accuracy.

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