Padé approximant related to the Wallis formula
AbstractBased on the Padé approximation method, in this paper we determine the coefficients aj$a_{j}$ and bj$b_{j}$ such that 
 π=((2n)!!(2n−1)!!)2{nk+a1nk−1+⋯+aknk+1+b1nk+⋯+bk+1+O(1n2k+3)},n→∞,$$ \pi= \biggl(\frac{(2n)!!}{(2n-1)!!} \biggr)^{2} \biggl\{ \frac {n^{k}+a_{1}n^{k-1}+\cdots+a_{k}}{n^{k+1}+b_{1}n^{k}+\cdots+b_{k+1}}+O \biggl(\frac{1}{n^{2k+3}} \biggr) \biggr\} ,\quad n\to\infty, $$ where k≥0$k\geq0$ is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π, which refines some known results.
Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function
Abstract In this paper, we present some completely monotonic functions and asymptotic expansions related to the gamma function. Based on the obtained expansions, we provide new bounds for Γ ( x + 1 ) / Γ x + 1 2 and Γ x + 1 2 .
Remarks on some completely monotonic functions
Applying the Euler–Maclaurin summation formula, we obtain upper and lower polynomial bounds for the function xex−1, x>0, with coefficients the Bernoulli numbers Bk. This enables us to give simpler proofs of some results of H. Alzer and F. Qi et al., concerning complete monotonicity of certain functions involving the gamma function Γ(x), the psi function ψ(x) and the polygamma functions ψ(n)(x), n=1,2,… .