On Some Asymptotic Expansions for the Gamma Function

Abstract

Inequalities play a fundamental role in both theoretical and applied mathematics and contain many patterns of symmetries. In many studies, inequalities have been used to provide estimates of some functions based on the properties of their symmetry. In this paper, we present the following new asymptotic expansion related to the ordinary gamma function $\Gamma (1+w)\sim \sqrt{2\pi w}{(w/e)}^w{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{w/2}exp\left({\sum }_{r=1}^{\infty }\frac{{\mu }_r}{w^r}\right),w\to \infty$, with the recurrence relation of coefficients ${\mu }_r$. Furthermore, we use Padé approximants and our new asymptotic expansion to deduce the new bounds of $\Gamma \left(w\right)$ better than some of its recent ones.

1. Introduction

Stirling’s formula

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(\frac{w}{e}\right)}^w,w\to \infty$$

(1)

is the most well known and used approximation formula for dealing with large factorials [1,2,3]. There are many researchers who have gone to great lengths to create more accurate approximations of $n!$ and its natural Gamma function extension $\Gamma \left(w\right)$, where $\Gamma \left(w\right)={\int }_0^{\infty }{e}^{-r}{r}^{w-1}dr$; $w>0$. For example, the Stirling series is stated as follows [4]:

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(\frac{w}{e}\right)}^{w}exp\left(\sum _{s=1}^{\infty }\frac{B_{2s}}{2s(2s-1)w^{2s-1}}\right),w\to \infty$$

(2)

which is an extension of Formula (1), where constants $B_{2s}$, $s\in \{0,1,2,...\}$, are Bernoulli numbers. Another asymptotic formula is given by Laplace [4].

$$\Gamma (1+w)\sim \sqrt{2\pi w}{\left(\frac{w}{e}\right)}^w\left(1+\frac{1}{12w}+\frac{1}{288w^2}-\frac{139}{51,840w^3}-\frac{571}{2,488,320w^4}+...\right),w\to \infty .$$

(3)

Burnside [5] presents the next most accurate approximation of Formula (1).

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(\frac{2w+1}{2e}\right)}^{w+1/2},w\to \infty .$$

(4)

Moreover, there are two important approximations that are better than Burnside’s Formula (4). The first one is due to Ramanujan [6]:

$$\Gamma (w+1)\sim \sqrt{\pi }{\left(\frac{w}{e}\right)}^w\sqrt[6]{8w^3+4w^2+w+\frac{1}{30}},w\to \infty$$

(5)

which presents a refinement of Stirling’s Formula (1) and was recorded in the book “The lost notebook and other unpublished papers” as a conjecture of Ramanujan based on some numerical calculations (see also [6,7,8,9,10]). The other one is due to Gosper [2].

$$\Gamma (w+1)\sim \sqrt{2\pi \left(w+1/6\right)}{\left(\frac{w}{e}\right)}^w,w\to \infty .$$

(6)

Starting from the Ramanujan Formula (5), Karatsuba presented [8] for $w\to \infty$ that

$$\Gamma (w+1)\sim \sqrt{\pi }{\left(w/e\right)}^w{\left[8w^3+4w^2+w+\frac{1}{30}-\frac{11}{240w}+\frac{79}{3360w^2}+\frac{3539}{201,600w^3}+...\right]}^{1/6}.$$

(7)

Mortici [11] improved the Ramanujan Formula (5) by the asymptotic formula:

$$\begin{array}{ccc}\hfill \Gamma (w+1)\sim & & \sqrt{\pi }{\left(w/e\right)}^w{\left[8w^3+4w^2+w+\frac{1}{30}\right]}^{1/6}exp\left[-\frac{11}{11520w^4}\right.\hfill \\ & & +\left.\frac{13}{3440w^5}+\frac{1}{691200w^6}+...\right],w\to \infty \hfill \end{array}$$

(8)

which is faster than Formula (7). In 2002, in a web post, Robert H. Windschitl pointed out that the following is the case [12] (see also [13]).

$$\Gamma (w+1)=\sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh{w}^{-1}\right)}^{w/2}\left(1+O\left(\frac{1}{w^5}\right)\right),w\to \infty .$$

(9)

Motivated by (9), Alzer [14] deduced the following double inequality:

$$\begin{array}{cc}\hfill \sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh{w}^{-1}\right)}^{w/2}\left(1+\frac{\theta }{w^5}\right)<\Gamma (w+1)& \\ \hfill <\sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh{w}^{-1}\right)}^{w/2}\left(1+\frac{\eta }{w^5}\right)& & ,w>0\hfill \end{array}$$

(10)

with $\theta =0$ and $\eta =\frac{1}{1620}$ being the best possible constants. Lu et al. [15] deduced the following extended formula to Windschitl’s formula:

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh\left(\frac{1}{w}+\frac{{\theta }_7}{w^7}+\frac{{\theta }_9}{w^9}+\frac{{\theta }_{11}}{w^{11}}+...\right)\right)}^{w/2},w\to \infty$$

(11)

where ${\theta }_7=1/810$, ${\theta }_9=-67/42525$, and ${\theta }_{11}=19/8505$. Chen [16] presented the following asymptotic expansion of the Gamma function related to Windschitl’s formula:

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh{w}^{-1}\right)}^{w/2+{\sum }_{k=0}^{\infty }\frac{{\tau }_k}{w^k}},w\to \infty$$

(12)

with a recurrence relation for determining coefficients ${\tau }_k$. Motivated by the above results, Yang and Tian [17] provided a more accurate Windschitl-type approximation:

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh{w}^{-1}\right)}^{w/2}{e}^{\frac{7}{324}\frac{1}{w^3(35w^2+33)}},w\to \infty .$$

(13)

They developed Windschitl’s approximation formula by giving two asymptotic expansions [18]:

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh{w}^{-1}\right)}^{w/2}exp\left(\sum _{k=3}^{\infty }\frac{s_k}{w^{2k-1}}\right),w\to \infty$$

(14)

where $s_k=\frac{2k(2k-2)!-2^{2k-1}}{2k\left(2k\right)!}{B}_{2k}$ and

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh{w}^{-1}\right)}^{w/2}\left(1+\sum _{k=1}^{\infty }\frac{{\delta }_k}{w^k}\right),w\to \infty$$

(15)

where ${\delta }_k=\frac{1}{k}{\sum }_{r=1}^k\left(\frac{1}{r+1}-\frac{2^r}{{(r+1)}^2(r-1)!}{B}_{r+1}{\delta }_{k-r}\right)$. Moreover, they [19] deduced the following family of high accurate approximation formulas for $q\ge -33/35$

$$\Gamma (w+1)\sim \sqrt{2\pi w}{\left(w/e\right)}^w{\left(wsinh{w}^{-1}\right)}^{w/2}exp\left(\frac{1}{1620w^5}\frac{w^2+q}{w^2+q+33/35}\right),w\to \infty .$$

(16)

Finally, they presented four new Windschitl-type approximation formulas [20]. Nemes [21] deduced that

$$\Gamma (w+1)=\sqrt{2\pi w}{\left(w/e\right)}^w{\left(1+\frac{1}{12w^2-1/10}\right)}^w\left(1+O\left(w^{-5}\right)\right),w\to \infty$$

(17)

which is much simpler than (9), and they have the same number of exact digits. Formulas (9) and (17) are stronger Ramanujan formulas. Starting from Nemes’s Formula (17), Mortici [22] constructed the continued fraction approximation:

$$\Gamma \left(w+1\right)\sim \sqrt{2\pi w}{e}^{-w}{\left(w+\frac{1}{12w+\frac{1}{10w+\frac{{\phi }_1}{w+\frac{{\phi }_2}{w+\frac{{\phi }_3}{w+\cdots }}}}}\right)}^w,$$

(18)

where ${\phi }_1=-\frac{2369}{252}$, ${\phi }_2=\frac{2117009}{1193976}$, and ${\phi }_3=\frac{393032191511}{1324011300744},\cdots$ . For more details about Gamma function applications, see [23,24] and the references therein.

For the rest of this paper, we will present two new asymptotic formulas for $\Gamma (w+1)$ depending on Windschitl (9) and Nemes’s (17) asymptotic formulas.

2. Lemmas

In sequel, the following results are required in our conclusions.

Lemma 1 

([25]). If we have the asymptotic expansion

$$h\left(w\right)\sim 1+\sum _{p=1}^{\infty }\frac{{\alpha }_p}{w^p},w\to \infty ,$$

then we obtain the following asymptotic expansion

$$lnh\left(w\right)\sim \sum _{p=1}^{\infty }\frac{{\beta }_p}{w^p},$$

where

$${\beta }_p={\alpha }_p-\frac{1}{p}\sum _{r=1}^{p-1}r{\beta }_r{\alpha }_{p-r},p\in \mathbb{N}.$$

Lemma 2 

([25]). If we have the asymptotic expansion

$$h\left(w\right)\sim \sum _{p=1}^{\infty }\frac{{\alpha }_p}{w^p},w\to \infty ,$$

then we obtain the following asymptotic expansion

$$exph\left(w\right)\sim 1+\sum _{p=1}^{\infty }\frac{{\beta }_p}{w^p},$$

where

$${\beta }_p=\frac{1}{p}\sum _{r=1}^{p}r{\beta }_{p-r}{\alpha }_r,p\in \mathbb{N}.$$

Lemma 3 

([26]). If ${\left\{{\zeta }_p\right\}}_{p\in \mathbb{N}}$ is a null sequence, $s\in \mathbb{R}$ and $n>1$ such that

$$\underset{p\to \infty }{lim}{p}^n({\zeta }_p-{\zeta }_{p+1})=s,$$

(19)

then we have

$$\underset{p\to \infty }{lim}{p}^{n-1}{\zeta }_p=\frac{s}{n-1}.$$

Lemma 4 

([27]). Let the real-valued function $\lambda \left(\rho \right)$ defined for $\rho >{\rho }_0$ and ${lim}_{\rho \to \infty }\lambda \left(\rho \right)=0$. If $\lambda (\rho +a)<\lambda \left(\rho \right)$, $a\in \mathbb{N}$, then $\lambda \left(\rho \right)>0$ for $\rho >{\rho }_0$; if $\lambda (\rho +a)>\lambda \left(\rho \right)$, $a\in \mathbb{N}$, then $\lambda \left(\rho \right)<0$ for $\rho >{\rho }_0$.

3. Main Results

To obtain the best possible constants $t_1$ and $t_2$ in the approximation formula

$$n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n{\left(\frac{n^2+t_1}{n^2+t_2}\right)}^{n/2},n\to \infty ,$$

(20)

we define a sequence $R_n$ that satisfies

$$n!=\sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n{\left(\frac{n^2+t_1}{n^2+t_2}\right)}^{n/2}{e}^{R_n},n\ge 1.$$

Then,

$$\begin{array}{ccc}\hfill R_n-R_{n+1}=& & \frac{-6t_1+6t_2+1}{12n^2}+\frac{6t_1-6t_2-1}{12n^3}+\frac{30t_1^2-20t_1-30t_2^2+20t_2+3}{40n^4}\hfill \\ & & +\frac{-45t_1^2+15t_1+45t_2^2-15t_2-2}{30n^5}\hfill \\ & & +\frac{-70t_1^3+210t_1^2-42t_1+70t_2^3-210t_2^2+42t_2+5}{84n^6}+O\left(n^{-13/2}\right).\hfill \end{array}$$

If $-6t_1+6t_2+1\ne 0$, then sequence $R_n-R_{n+1}$ has a rate of worse than $n^{-2}$. So, we will consider $t_2=t_1-\frac{1}{6}$. Moreover, to increase the rate of convergence, we use $30t_1^2-20t_1-30t_2^2+20t_2+3=0$ and then $t_1=\frac{7}{60}$ and $t_2=-\frac{1}{20}$. Now, by Lemma 3, we obtain the following result.

Lemma 5. 

The sequence is as follows:

$$R_n=ln\left(\frac{n!{\left(\frac{n^2+\frac{7}{60}}{n^2-\frac{1}{20}}\right)}^{-n/2}}{\sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n}\right)=O\left(n^{-5}\right),n\to \infty$$

(21)

where

$$\underset{n\to \infty }{lim}\frac{R_n-R_{n+1}}{n^{-6}}=\frac{461}{181440}.$$

Theorem 1. 

The Gamma function has the following asymptotic expansion:

$$\Gamma (w+1)\sim \sqrt{2\pi w}{(w/e)}^w{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{w/2}exp\left(\sum _{r=1}^{\infty }\frac{{\mu }_r}{w^r}\right),w\to \infty$$

(22)

where ${\mu }_r$ are given by

$${\mu }_r=\frac{B_{r+1}}{r(r+1)}-\frac{1}{2}{\nu }_{r+1},r=1,2,3,...$$

(23)

where $B_r$ denote Bernoulli numbers and

$${\nu }_r={\sigma }_r-\frac{1}{r}\sum _{j=1}^{r-1}j{\nu }_j{\sigma }_{r-j},r=1,2,3,...$$

with

$${\sigma }_0=1,{\sigma }_{2s}=\frac{10}{3{\left(20\right)}^s},{\sigma }_{2s-1}=0,s=1,2,3,...$$

and the empty sum is understood as usual to be nil.

Proof. 

From Stirling series (2) with the identity $B_{2r+1}=0$, $r=1,2,3,...$, we have the following.

$$ln\left(\frac{\Gamma (w+1)}{\sqrt{2\pi w}{\left(\frac{w}{e}\right)}^w}\right)\sim \sum _{r=1}^{\infty }\frac{B_{r+1}}{r(r+1)}{w}^{-r},w\to \infty .$$

(24)

Moreover, we obtain

$$\begin{array}{ccc}\hfill \frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}& =& 1+\sum _{n=1}^{\infty }\frac{10}{3{\left(20\right)}^n}{\left(w^2\right)}^{-n}\hfill \\ \hfill & =& \sum _{s=0}^{\infty }{\sigma }_s{w}^{-s},\left|w\right|>\sqrt{\frac{1}{20}}\hfill \end{array}$$

with

$${\sigma }_0=1,{\sigma }_{2s}=\frac{10}{3{\left(20\right)}^s},{\sigma }_{2s-1}=0,s=1,2,3,....$$

By Lemma 1, we obtain

$$ln\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)\sim \sum _{r=1}^{\infty }{\nu }_r{w}^{-r},w\to \infty$$

(25)

with

$${\nu }_r={\sigma }_r-\sum _{j=1}^{r-1}\frac{j{\nu }_j{\sigma }_{r-j}}{r},r=1,2,3,....$$

Hence,

$$\sum _{r=1}^{\infty }\frac{B_{r+1}}{r(r+1)w^r}-\sum _{r=1}^{\infty }\frac{{\nu }_r}{2w^{r-1}}\sim \sum _{r=1}^{\infty }\frac{{\mu }_r}{w^r},w\to \infty$$

which gives us with the aid of $v_1=0$ such that

$$\sum _{r=1}^{\infty }\left(\frac{B_{r+1}}{r(r+1)}-\frac{{\nu }_{r+1}}{2}\right)w^{-r}\sim \sum _{r=1}^{\infty }{\mu }_r{w}^{-r},w\to \infty .$$

(26)

Equating coefficients of $w^{-r}$ yields

$${\mu }_r=\frac{B_{r+1}}{r(r+1)}-\frac{{\nu }_{r+1}}{2},r=1,2,3,....$$

□

Remark 1. 

According to Theorem 1, we have the following explicit formula.

$$\begin{array}{ccc}\hfill \Gamma (w+1)\sim & & \sqrt{2\pi w}{(w/e)}^w{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{w/2}exp\left(\frac{461}{907200w^5}-\frac{5197}{9072000w^7}\right.\hfill \\ & & +\frac{1436249}{1710720000w^9}-\frac{26863154077}{14010796800000w^{11}}+\frac{326590926551}{50948352000000w^{13}}\hfill \\ & & \left.-\frac{8437736988187}{285534720000000w^{15}}+...\right),w\to \infty .\hfill \end{array}$$

(27)

Using Lemma 2 and Theorem 1, we obtain the following result.

Theorem 2. 

The Gamma function has the asymptotic expansion

$$\Gamma (w+1)\sim \sqrt{2\pi w}{(w/e)}^w{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{w/2}\left(\sum _{r=0}^{\infty }\frac{{\lambda }_r}{w^r}\right),w\to \infty$$

(28)

where ${\lambda }_r$ are given by

$${\lambda }_0=1,{\lambda }_r=\frac{1}{r}\sum _{j=1}^{r}j{\lambda }_{r-j}{\mu }_j,r=1,2,3,...$$

(29)

with coefficients ${\mu }_r$ being given by (23).

Remark 2. 

According to Theorem 2, we have

$$\Gamma (w+1)\sim \sqrt{2\pi w}{(w/e)}^w{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{w/2}\left(1+O\left(w^{-5}\right)\right),w\to \infty$$

(30)

which is simpler than Windschitl’s approximation (9) and has the same rate of convergence.

4. Numerical Comparisons among Some Approximation Formulas of $\mathit{n}!$

We have the following approximation formulas:

$n!\sim$	$\sqrt{2\pi n}{\left(n/e\right)}^n{\left(nsinh\left(\frac{1}{n}+\frac{1}{810n^7}\right)\right)}^{n/2}$	$\doteqdot a_n,$	[15]
$n!\sim$	$\sqrt{2\pi n}{\left(n/e\right)}^n{\left(1+\frac{1}{12-\frac{1}{10n^2-\frac{2369}{252}}}\right)}^n$	$\doteqdot b_n,$	[22]
$n!\sim$	$\sqrt{2\pi }{(n+1)}^{n+1/2}{e}^{-n-1}exp\left(\frac{1}{12n}-\frac{1}{12n^2}+\frac{29}{360n^3}-\frac{3}{40n}\right)$	$\doteqdot c_n,$	[28]
$n!\sim$	$\sqrt{2\pi }{(n+1)}^n{e}^{-n-1/2}exp{\left(n^3+5/4n^2+17/32n+\frac{172}{1920}\right)}^{1/6}$	$\doteqdot d_n,$	[29]
$n!\sim$	$\sqrt{2\pi n}{\left(n/e\right)}^n{\left(\frac{n^2-9/120}{n^2-1/120}\right)}^n$	$\doteqdot f_n,$	[21]

and

$n!\sim$

$\sqrt{2\pi n}{\left(n/e\right)}^n{\left(nsinh\frac{1}{n}\right)}^{n/2}exp\left(\frac{n}{2}+\frac{1}{270n^3}\right)$

$\doteqdot g_n,$

[16].

Chen [16] showed by some numerical computations that approximation $g_n$ is stronger than the others. Moreover, Chen [30] showed by some numerical computations that the two approximations

$$n!\sim \sqrt{2\pi n}{\left(n/e\right)}^n{\left(nsinh{n}^{-1}\right)}^{n/2}exp\left(\frac{1}{1620}-\frac{11}{18900n^7}\right)\doteqdot h_n$$

and

$$n!\sim \sqrt{2\pi n}{\left(n/e\right)}^n{\left(nsinh{n}^{-1}\right)}^{n/2}\left(1+\frac{1}{1620}-\frac{11}{18900n^7}\right)\doteqdot k_n$$

are stronger than formula $g_n$ for $n\ge 2$. Table 1 shows that the approximation

$$n!\sim \sqrt{2\pi n}{(n/e)}^n{\left(\frac{n^2+\frac{7}{60}}{n^2-\frac{1}{20}}\right)}^{n/2}exp\left(\frac{461}{907200n^5}-\frac{5197}{9072000n^7}\right)\doteqdot l_n$$

is stronger than approximation $h_n$.

Table 1. Comparison among $h_n$ and $l_n$.

n	$\left|\frac{\mathit{n}!-{\mathit{h}}_{\mathit{n}}}{\mathit{n}!}\right|$	$\left|\frac{\mathit{n}!-{\mathit{l}}_{\mathit{n}}}{\mathit{n}!}\right|$
1	0.000306466	0.000305449
10	$8.22120\times {10}^{-13}$	$8.20998\times {10}^{-13}$
50	$4.30037\times {10}^{-19}$	$4.29462\times {10}^{-19}$
100	$8.40490\times {10}^{-22}$	$8.39367\times {10}^{-22}$
1000	$8.40680\times {10}^{-31}$	$8.39556\times {10}^{-31}$

Moreover, Table 2 shows that approximation

$$n!\sim \sqrt{2\pi n}{(n/e)}^n{\left(\frac{n^2+\frac{7}{60}}{n^2-\frac{1}{20}}\right)}^{n/2}\left(1+\frac{461}{907200n^5}-\frac{5197}{9072000n^7}\right)\doteqdot p_n$$

is stronger than approximation $k_n$:

Table 2. Comparison among $k_n$ and $p_n$.

n	$\left|\frac{\mathit{n}!-{\mathit{k}}_{\mathit{n}}}{\mathit{n}!}\right|$	$\left|\frac{\mathit{n}!-{\mathit{p}}_{\mathit{n}}}{\mathit{n}!}\right|$
1	0.000306466	0.000305451
10	$8.22139\times {10}^{-13}$	$8.21011\times {10}^{-13}$
50	$4.30039\times {10}^{-19}$	$4.29463\times {10}^{-19}$
100	$8.40492\times {10}^{-22}$	$8.39368\times {10}^{-22}$
1000	$8.40680\times {10}^{-31}$	$8.39556\times {10}^{-31}$

5. Some Bounds of $\Gamma (\mathit{w}+\mathbf{1})$ Using Padé Approximants

Consider the following formal power series.

$$K\left(w\right)={\alpha }_0+{\alpha }_{1}w+{\alpha }_2{w}^2+...,$$

Then, the rational function

$${[m,s]}_K\left(w\right)=\frac{{\displaystyle \sum _{i=0}^m}{\gamma }_i{w}^i}{1+{\displaystyle \sum _{i=1}^s}{\beta }_i{w}^i},m\ge 0;s\ge 1$$

(31)

is called the Padé approximant of order $(m,s)$ of function $K\left(w\right)$ [31,32,33], where

$${[m,s]}_K\left(w\right)-K\left(w\right)=O\left(w^{m+s+1}\right),w\to \infty$$

(32)

and coefficients ${\beta }_i$ are the solutions of the system

$$\left[\begin{array}{c}0\\ 0\\ ⋮\\ 0\end{array}\right]=\left[\begin{array}{cccc}{\alpha }_{m+1}& {\alpha }_m& ...& {\alpha }_{m-s+1}\\ {\alpha }_{m+2}& {\alpha }_{m+1}& ...& {\alpha }_{m-s+2}\\ ⋮& ⋮& ⋮& ⋮\\ {\alpha }_{m+s}& {\alpha }_{m+s-1}& ...& {\alpha }_m\end{array}\right]\left[\begin{array}{c}1\\ {\beta }_1\\ ⋮\\ {\beta }_s\end{array}\right]$$

(33)

with ${\alpha }_j=0$ for $j<0$, and coefficients ${\gamma }_i$ are given by the following.

$$\left.\begin{array}{ccc}\hfill {\gamma }_0& =& {\alpha }_0\hfill \\ \hfill {\gamma }_1& =& {\alpha }_1+{\alpha }_0{\beta }_1\hfill \\ \hfill & ⋮& \\ \hfill {\gamma }_m& =& {\alpha }_m+{\alpha }_{m-1}{\beta }_1+\cdots +{\alpha }_{m-s}{\beta }_s\hfill \end{array}\right\}.$$

(34)

For the formal power series $f\left(w\right)\doteqdot {\sum }_{r=1}^{\infty }\frac{{\mu }_r}{w^r}$, where ${\mu }_r$ are given by (23), we can conclude the following Padé approximations

$${[5,5]}_f\left(w\right)=\frac{461}{907200\left(-\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}+O\left(w^{-11}\right)$$

and

$${[5,6]}_f\left(w\right)=\frac{461}{907200\left(\frac{89617471668379}{60523294534560w^6}-\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}+O\left(w^{-12}\right).$$

Hence, we obtain the following double inequality.

Theorem 3. 

The following double inequality holds for $w\ge 1$.

$$exp\left(\frac{461}{907200\left(\frac{89617471668379}{60523294534560w^6}-\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}\right)$$

$$<\frac{\Gamma (w+1)}{\sqrt{2\pi w}{(w/e)}^w}{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{-w/2}<$$

$$exp\left(\frac{461}{907200\left(-\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}\right).$$

(35)

Proof. 

Consider the function

$$M\left(w\right)=\frac{exp\left(-\frac{461}{907200\left(\frac{89617471668379}{60523294534560w^6}-\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}\right)\Gamma (w+1)}{\sqrt{2\pi w}{\left(\frac{w}{e}\right)}^w{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{w/2}},w\ge 1$$

and then

$$M^{\prime }\left(w\right)=\frac{e^{\frac{w\left(12709891852257600w^6+14328266367935520w^4-4846102975366380w^2+18813210430271527\right)}{210\left(60523294534560w^6+68229839847312w^4-23076680835078w^2+89617471668379\right)}}\Gamma (w+1)}{\sqrt{2\pi }{w}^{w+\frac{1}{2}}{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{\frac{w}{2}}}{M}_1\left(w\right),$$

where

$$\begin{array}{ccc}\hfill M_1\left(w\right)& =& \frac{1}{20w^2-1}+\frac{7}{60w^2+7}-\frac{1}{2w}-\frac{1}{2}log\left(\frac{10}{60w^2-3}+1\right)-log\left(w\right)+\psi (w+1)\hfill \\ & -& \frac{6458620088063\left(71916\left(316247844w^2-213922547\right)w^2+89617471668379\right)}{35{\left(107874\left(121704w^2\left(4610w^2+5197\right)-213922547\right)w^2+89617471668379\right)}^2}\hfill \\ & +& \frac{6458620088063}{42\left(107874\left(121704w^2\left(4610w^2+5197\right)-213922547\right)w^2+89617471668379\right)}.\hfill \end{array}$$

Using $M_2\left(w\right)=M_1(w+1)-M_1\left(w\right)$, we obtain

$$M_2^{\prime }(w+1)=\frac{M_3\left(w\right)}{M_4\left(w\right)}<0,w\ge 0$$

where

$$\begin{array}{ccc}\hfill M_3\left(w\right)& =& -6.0\times {10}^{97}{w}^{40}-3.6\times {10}^{99}{w}^{39}-1.1\times {10}^{101}{w}^{38}-2.0\times {10}^{102}{w}^{37}-2.8\times {10}^{103}{w}^{36}\hfill \\ & & -3.0\times {10}^{104}{w}^{35}-2.6\times {10}^{105}{w}^{34}-1.9\times {10}^{106}{w}^{33}-1.2\times {10}^{107}{w}^{32}-6.3\times {10}^{107}{w}^{31}\hfill \\ & & -2.9\times {10}^{108}{w}^{30}-1.2\times {10}^{109}{w}^{29}-4.3\times {10}^{109}{w}^{28}-1.4\times {10}^{110}{w}^{27}-4.0\times {10}^{110}{w}^{26}\hfill \\ & & -1.0\times {10}^{111}{w}^{25}-2.4\times {10}^{111}{w}^{24}-5.1\times {10}^{111}{w}^{23}-9.8\times {10}^{111}{w}^{22}-1.7\times {10}^{112}{w}^{21}\hfill \\ & & -2.7\times {10}^{112}{w}^{20}-3.8\times {10}^{112}{w}^{19}-4.9\times {10}^{112}{w}^{18}-5.7\times {10}^{112}{w}^{17}-6.0\times {10}^{112}{w}^{16}\hfill \\ & & -5.7\times {10}^{112}{w}^{15}-4.9\times {10}^{112}{w}^{14}-3.8\times {10}^{112}{w}^{13}-2.6\times {10}^{112}{w}^{12}-1.6\times {10}^{112}{w}^{11}\hfill \\ & & -8.9\times {10}^{111}{w}^{10}-4.3\times {10}^{111}{w}^9-1.8\times {10}^{111}{w}^8-6.6\times {10}^{110}{w}^7-2.0\times {10}^{110}{w}^6\hfill \\ & & -5.3\times {10}^{109}{w}^5-1.1\times {10}^{109}{w}^4-1.8\times {10}^{108}{w}^3-2.2\times {10}^{107}{w}^2-1.7\times {10}^{106}w\hfill \\ & & -6.7\times {10}^{104}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill M_4\left(w\right)& =& 70{(w+2)}^2{\left(20w^2+40w+19\right)}^2{\left(20w^2+80w+79\right)}^2{\left(60w^2+120w+67\right)}^2\hfill \\ & & {\left(60w^2+240w+247\right)}^2{(w+1)}^2(60523294534560w^6+363139767207360w^5\hfill \\ & & +976079257865712w^4+1483385250080448w^3+1294151776267194w^2\hfill \\ & & {+589905764926452w+195293925215173)}^3(60523294534560w^6\hfill \\ & & +726279534414720w^5+3699627511920912w^4+10229565844308096w^3\hfill \\ & & +16140030163794810w^2{+13711520702409192w+4962479036096899)}^3.\hfill \end{array}$$

Now, $M_2\left(w\right)$ is a decreasing function for $w\ge 1$ with ${lim}_{w\to \infty }{M}_2\left(w\right)=0$; then, $M_2\left(w\right)>0$ or $M_1\left(w\right)<M_1(w+1)$ for $w\ge 1$. However, ${lim}_{w\to \infty }{M}_1\left(w\right)=0$; hence, by using Lemma 4, we have $M_1\left(w\right)<0$ for $w\ge 1$. Then, $M\left(w\right)$ is a decreasing function for $w\ge 1$ with ${lim}_{w\to \infty }M\left(w\right)=1$ and then $M\left(w\right)>1$ for $w\ge 1$ or

$$\frac{\Gamma (w+1)}{\sqrt{2\pi w}{\left(\frac{w}{e}\right)}^w{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{w/2}}>exp\left(\frac{461}{907200\left(\frac{89617471668379}{60523294534560w^6}-\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}\right),w\ge 1.$$

Now, consider the following function

$$H\left(w\right)=\frac{e^{w-\frac{461}{907200\left(-\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}}{w}^{-w-\frac{1}{2}}{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{-\frac{w}{2}}\Gamma (w+1)}{\sqrt{2\pi }},$$

and then

$$H^{\prime }\left(w\right)=\frac{e^{w-\frac{461}{907200\left(-\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}}{w}^{-w-\frac{1}{2}}{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{-\frac{w}{2}}\Gamma (w+1)}{\sqrt{2\pi }}{H}_1\left(w\right),$$

where

$$\begin{array}{ccc}\hfill H_1\left(w\right)& =& \frac{200w^2}{1200w^4+80w^2-7}-\frac{1}{2w}-\frac{1}{2}log\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)-log\left(w\right)+\psi (w+1)\hfill \\ & & -\frac{1077693991\left(316247844w^2-213922547\right)}{945w^2{\left(561055440w^4+632495688w^2-213922547\right)}^2}\hfill \\ & & +\frac{1077693991}{756w^2\left(561055440w^4+632495688w^2-213922547\right)}.\hfill \end{array}$$

Using $H_2\left(w\right)=H_1(w+1)-H_1\left(w\right)$, we obtain

$$H_2^{\prime }(w+1)=\frac{H_3\left(w\right)}{H_4\left(w\right)}<0,w\ge 0$$

where

$$\begin{array}{ccc}\hfill H_3\left(w\right)& =& 1.6\times {10}^{68}{w}^{32}+7.6\times {10}^{69}{w}^{31}+1.8\times {10}^{71}{w}^{30}+2.6\times {10}^{72}{w}^{29}+2.9\times {10}^{73}{w}^{28}\hfill \\ & & +2.4\times {10}^{74}{w}^{27}+1.6\times {10}^{75}{w}^{26}+9.1\times {10}^{75}{w}^{25}+4.2\times {10}^{76}{w}^{24}+1.7\times {10}^{77}{w}^{23}\hfill \\ & & +5.8\times {10}^{77}{w}^{22}+1.8\times {10}^{78}{w}^{21}+4.6\times {10}^{78}{w}^{20}+1.1\times {10}^{79}{w}^{19}+2.1\times {10}^{79}{w}^{18}\hfill \\ & & +3.8\times {10}^{79}{w}^{17}+6.1\times {10}^{79}{w}^{16}+8.6\times {10}^{79}{w}^{15}+1.1\times {10}^{80}{w}^{14}+1.2\times {10}^{80}{w}^{13}\hfill \\ & & +1.1\times {10}^{80}{w}^{12}+9.6\times {10}^{79}{w}^{11}+7.1\times {10}^{79}{w}^{10}+4.6\times {10}^{79}{w}^9+2.5\times {10}^{79}{w}^8\hfill \\ & & +1.2\times {10}^{79}{w}^7+4.7\times {10}^{78}{w}^6+1.5\times {10}^{78}{w}^5+3.9\times {10}^{77}{w}^4+7.8\times {10}^{76}{w}^3\hfill \\ & & +1.1\times {10}^{76}{w}^2+1.0\times {10}^{75}w+4.6\times {10}^{73}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill H_4\left(w\right)& =& {\left(561055440w^4+2244221760w^3+3998828328w^2+3509213136w+979628581\right)}^3\hfill \\ & & {\left(561055440w^4+4488443520w^3+14097826248w^2+20483756832w+11292947245\right)}^3\hfill \\ & & 1890{(w+1)}^3{(w+2)}^3{\left(20w^2+40w+19\right)}^2{\left(20w^2+80w+79\right)}^2{\left(60w^2+120w+67\right)}^2\hfill \\ & & {\left(60w^2+240w+247\right)}^2.\hfill \end{array}$$

Now, $H_2\left(w\right)$ is an increasing function for $w\ge 1$ with ${lim}_{w\to \infty }{H}_2\left(w\right)=0$; then, $H_2\left(w\right)<0$ or $H_1(w+1)<H_1\left(w\right)$ for $w\ge 1$. However, ${lim}_{w\to \infty }{H}_1\left(w\right)=0$; hence, using Lemma 4, we have $H_1\left(w\right)>0$ for $w\ge 1$. Then, $H\left(w\right)$ is an increasing function for $w\ge 1$ with ${lim}_{w\to \infty }H\left(w\right)=1$ and then $H\left(w\right)<1$ for $w\ge 1$ or

$$\frac{\Gamma (w+1)}{\sqrt{2\pi w}{\left(\frac{w}{e}\right)}^w{\left(\frac{w^2+\frac{7}{60}}{w^2-\frac{1}{20}}\right)}^{w/2}}<e^{\frac{461}{907200\left(\frac{213922547}{561055440w^4}+\frac{5197}{4610w^2}+1\right)w^5}},w\ge 1.$$

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Remark 3. 

The bounds in our new inequality (35) are better than the bounds in Alzer’s inequality (10) for $w\ge 1$.

6. Conclusions

The Padé approximant method and asymptotic expansions can be employed to present some new bounds of $\Gamma \left(w\right)$. We presented proofs to illustrate the novelty of our findings, which may be of interest to a significant portion of the readers. This method is a powerful tool for inferring inequalities for many other special functions.