Padé approximant related to asymptotics for the gamma function

Abstract

Abstract Based on the Padé approximation method, we determine the coefficients a j $a_{j}$ and b j $b_{j}$ ( 1 ≤ j ≤ k $1\leq j\leq k$ ) such that Γ ( x + 1 ) 2 π x ( x / e ) x = x k + a 1 x k − 1 + ⋯ + a k x k + b 1 x k − 1 + ⋯ + b k + O ( 1 x 2 k + 1 ) , x → ∞ , $$ \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x^{k}+a_{1}x^{k-1} +\cdots+a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots+b_{k}}+O \biggl(\frac {1}{x^{2k+1}} \biggr),\quad x\to \infty, $$ where k ≥ 1 $k\geq1$ is any given integer. Based on the obtained result, we establish new bounds for the gamma function.

Keywords

• gamma function
• psi function
• inequality
• approximation