Padé approximant related to asymptotics for the gamma function

Based on the Padé approximation method, we determine the coefficients aj\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} and bj\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} (1≤j≤k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq j\leq k$\end{document}) such that Γ(x+1)2πx(x/e)x=xk+a1xk−1+⋯+akxk+b1xk−1+⋯+bk+O(1x2k+1),x→∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\Gamma(x+1)}{\sqrt{2\pi x}(x/e)^{x}}=\frac{x^{k}+a_{1}x^{k-1} +\cdots+a_{k}}{x^{k}+b_{1}x^{k-1}+\cdots+b_{k}}+O \biggl(\frac {1}{x^{2k+1}} \biggr),\quad x\to \infty, $$\end{document} where k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k\geq1$\end{document} is any given integer. Based on the obtained result, we establish new bounds for the gamma function.


Introduction
Stirling's formula The following asymptotic formula is due to Laplace: as x → ∞ (see [], p., Eq. (..)). The expression (.) is sometimes incorrectly called Stirling's series (see [], pp.-). Stirling's formula is in fact the first approximation to the asymptotic formula (.). Stirling's formula has attracted much interest of many mathematicians and has motivated a large number of research papers concerning various generalizations and improvements (see [-] and the references cited therein). It is interesting to note that the aforementioned mathematicians represent many nationalities. So the topic is of interest for mathematicians from diverse cultural background.
Using the Maple software, we find, as x → ∞, and Based on the Padé approximation method, in this paper we develop the approximation formulas (.) and (.) to produce a general result. More precisely, we determine the coefficients a j and b j ( ≤ j ≤ k) such that where k ≥  is any given integer. Based on the obtained result, we establish new bounds for the gamma function.
The numerical values given in this paper have been calculated via the computer program MAPLE .

Lemmas
The following lemmas are required in our present investigation.
Lemma . ([]) Let r be a given nonzero real number. The gamma function has the following asymptotic formula: with the coefficients b j = b j (r) (j = , , . . .) given by where B n (n ∈ N  := N ∪ {}) are the Bernoulli numbers defined in (.), summed over all nonnegative integers k j satisfying the equation k  + k  + · · · + jk j = j.
Laplace formula (.) can be rewritten as with the coefficients c j given by Remark . Lemma . can be stated as follows: for every m ∈ N  , the function In , Koumandos [] presented a simpler proof of complete monotonicity of the functions R m (x). In , Koumandos and Pedersen [], Theorem ., strengthened this result.
From F n (x) <  and G n (x) <  for x > , we obtain

Approximations to the gamma function
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 [p/q] f are identical to those of f . Moreover, we have (see []) with f n (x) = c  + c  x + · · · + c n x n , the nth partial sum of the series f in (.) (f n is identically zero for n < ). 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 holds, we have, by (.), We thus obtain that and we have, by (.), We now give a derivation of formula (.). To this end, we consider Noting that (.) holds, we have, by (.), that is, We thus obtain that and we have, by (.), From the Padé approximation method and the expansion (.), we now present a general result given by Theorem ..
where p ≥  and q ≥  are any given integers, the coefficients a j and b j are given by and c j is given in (.), and the following holds: (.) Remark . Using (.), we can also derive (.) and (.). Indeed, we have Setting (p, q) = (k, k) in (.), we obtain the following corollary.

Corollary . As x → ∞,
where k ≥  is any given integer, the coefficients a j and b j ( ≤ j ≤ k) are given by and c j is given in (.).
Conjecture . The coefficients a j and b j ( ≤ j ≤ k) in (.) satisfy the following relation:

Inequalities for the gamma function
Formulas (.) and (.) motivate us to establish the following theorem.
Theorem . The following inequalities hold: The left-hand side inequality holds for x ≥ , while the right-hand side inequality is valid for x ≥ .
Proof It suffices to show that and Differentiating F(x) and applying the second inequality in (.) yield Hence, F (x) <  for x ≥ , and we have Differentiating G(x) and applying the first inequality in (.) yield Hence, G (x) >  for x ≥ , and we have The proof is complete.
Remark . Following the same method as the one used in the proof of Theorem ., we can prove the double inequality for x ≥ . We here omit it. Some computer experiments indicate that inequalities (.) and (.) are valid for x ≥ .
In view of (.) and (.), we pose the following conjecture.
Conjecture . If k is odd, then for x ≥ , where the coefficients a j and b j ( ≤ j ≤ k) are determined in (.). If k is even, then inequality (.) is reversed.

Comparison
In , Mortici   We here offer some numerical computations (see Table ) to show the superiority of our sequences (U n ) n≥ and (V n ) n≥ over the sequence (λ n ) n≥ .