Padé approximant related to the Wallis formula

Based on the Padé approximation method, in this paper we determine the coefficients \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{j}$\end{document}aj and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b_{j}$\end{document}bj such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \pi= \biggl(\frac{(2n)!!}{(2n-1)!!} \biggr)^{2} \biggl\{ \frac {n^{k}+a_{1}n^{k-1}+\cdots+a_{k}}{n^{k+1}+b_{1}n^{k}+\cdots+b_{k+1}}+O \biggl(\frac{1}{n^{2k+3}} \biggr) \biggr\} ,\quad n\to\infty, $$\end{document}π=((2n)!!(2n−1)!!)2{nk+a1nk−1+⋯+aknk+1+b1nk+⋯+bk+1+O(1n2k+3)},n→∞, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k\geq0$\end{document}k≥0 is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π, which refines some known results.

This result is due to Wallis (see []).
Based on a basic theorem in mathematical statistics concerning unbiased estimators with minimum variance, Gurland [] yielded a closer approximation to π than that af- We see from (.) that Based on the Padé approximation method, in this paper we develop the approximation formula (.) to produce a general result. More precisely, we determine the coefficients a j and b j such that where k ≥  is any given integer. Based on the obtained result, we establish a more accurate formula for approximating π , which refines some known results.
The numerical values given in this paper have been calculated via the computer program MAPLE .

Lemmas
Euler's gamma function (x) is one of the most important functions in mathematical analysis and has applications in diverse areas. The logarithmic derivative of (x), denoted by ψ(x) = (x)/ (x), is called the psi (or digamma) function.
The following lemmas are required in the sequel.

Lemma . ([])
Let r =  be a given real number and ≥  be a given integer. The following asymptotic expansion holds: with the coefficients p j ≡ p j ( , r) (j ∈ N) given by where B j are the Bernoulli numbers summed over all nonnegative integers k j satisfying the equation ( + )k  + ( + )k  + · · · + (j + -)k j = j.
In particular, setting ( , r) = (, -) in (.) yields where the coefficients c j ≡ p j (, -) (j ∈ N) are given by summed over all nonnegative integers k j satisfying the equation In particular, we have For our later use, we introduce Padé approximant (see [-]). Let f be a formal power series The Padé approximation of order (p, q) of the function f is the rational function, denoted by where p ≥  and q ≥  are two given integers, the coefficients a j and b j are given by (see and the following holds: Thus, the first p + q +  coefficients of the series expansion of with f n (x) = c  + c  x + · · · + c n x n , the nth partial sum of the series f in (.).

Main results
Let with the coefficients c j given by (.). In what follows, the function f is given in (.).
Based on the Padé approximation method, we now give a derivation of formula (.). To this end, we consider Noting that We thus obtain that and we have, by (.), Noting that holds, replacing x by n in (.) yields (.). From the Padé approximation method introduced in Section  and the asymptotic expansion (.), we obtain a general result given by Theorem .. As a consequence, we obtain (.).
Theorem . The Padé approximation of order (p, q) of the asymptotic formula of the (x+) )  (at the point x = ∞) is the following rational function: where p ≥  and q ≥  are two given integers and q = p +  (an empty sum is understood to be zero), the coefficients a j and b j are given by and c j is given in (.), and the following holds: Moreover, we have  Remark . Using (.), we can also derive (.). Indeed, we have Replacing x by n in (.) applying (.), we obtain the following corollary.
The left-hand side inequality holds for x ≥ , while the right-hand side inequality is valid for x ≥ .
Proof It suffices to show that Using the following asymptotic expansion (see []): Differentiating F(x) and applying the first inequality in (.), we find Hence, F (x) <  for x ≥ , and we have Differentiating G(x) and applying the second inequality in (.), we find Hence, G (x) >  for x ≥ , and we have The proof is complete.
The following numerical computations (see Table ) would show that δ n < a n and b n < ω n for n ∈ N. That is to say, inequalities (.) are sharper than inequalities (.).
In fact, we have