On Some Asymptotic Expansions for the Gamma Function

Mahmoud, Mansour; Almuashi, Hanan

doi:10.3390/sym14112459

Open AccessArticle

On Some Asymptotic Expansions for the Gamma Function

by

Mansour Mahmoud

^1,* and

Hanan Almuashi

²

¹

Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²

Mathematics Department, Faculty of Science, University of Jeddah, P.O. Box 80327, Jeddah 21589, Saudi Arabia

^*

Author to whom correspondence should be addressed.

Symmetry 2022, 14(11), 2459; https://doi.org/10.3390/sym14112459

Submission received: 8 November 2022 / Revised: 15 November 2022 / Accepted: 18 November 2022 / Published: 20 November 2022

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials)

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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

Γ (1 + w) \sim \sqrt{2 π w} {(w / e)}^{w} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{w / 2} exp (\sum_{r = 1}^{\infty} \frac{μ_{r}}{w^{r}}), w \to \infty

, with the recurrence relation of coefficients

μ_{r}

. Furthermore, we use Padé approximants and our new asymptotic expansion to deduce the new bounds of

Γ (w)

better than some of its recent ones.

Keywords:

asymptotic expansion; Stirling’s formula; gamma function; Windschitl’s formula; Padé approximants

MSC:

33B15; 41A60; 41A21

1. Introduction

Stirling’s formula

Γ (w + 1) \sim \sqrt{2 π w} {(\frac{w}{e})}^{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

Γ (w)

, where

Γ (w) = \int_{0}^{\infty} e^{- r} r^{w - 1} d r

;

w > 0

. For example, the Stirling series is stated as follows [4]:

Γ (w + 1) \sim \sqrt{2 π w} {(\frac{w}{e})}^{w} exp (\sum_{s = 1}^{\infty} \frac{B_{2 s}}{2 s (2 s - 1) w^{2 s - 1}}), w \to \infty

(2)

which is an extension of Formula (1), where constants

B_{2 s}

,

s \in {0, 1, 2, . . .}

, are Bernoulli numbers. Another asymptotic formula is given by Laplace [4].

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

(3)

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

Γ (w + 1) \sim \sqrt{2 π w} {(\frac{2 w + 1}{2 e})}^{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]:

Γ (w + 1) \sim \sqrt{π} {(\frac{w}{e})}^{w} \sqrt[6]{8 w^{3} + 4 w^{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].

Γ (w + 1) \sim \sqrt{2 π (w + 1 / 6)} {(\frac{w}{e})}^{w}, w \to \infty .

(6)

Starting from the Ramanujan Formula (5), Karatsuba presented [8] for

w \to \infty

that

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

(7)

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

\begin{matrix} Γ (w + 1) \sim & \sqrt{π} {(w / e)}^{w} {[8 w^{3} + 4 w^{2} + w + \frac{1}{30}]}^{1 / 6} exp [- \frac{11}{11520 w^{4}} \\ + \frac{13}{3440 w^{5}} + \frac{1}{691200 w^{6}} + . . .], w \to \infty \end{matrix}

(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]).

Γ (w + 1) = \sqrt{2 π w} {(w / e)}^{w} {(w sinh w^{- 1})}^{w / 2} (1 + O (\frac{1}{w^{5}})), w \to \infty .

(9)

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

\begin{matrix} \sqrt{2 π w} {(w / e)}^{w} {(w sinh w^{- 1})}^{w / 2} (1 + \frac{θ}{w^{5}}) < Γ (w + 1) \\ < \sqrt{2 π w} {(w / e)}^{w} {(w sinh w^{- 1})}^{w / 2} (1 + \frac{η}{w^{5}}) & , w > 0 \end{matrix}

(10)

with

θ = 0

and

η = \frac{1}{1620}

being the best possible constants. Lu et al. [15] deduced the following extended formula to Windschitl’s formula:

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(w sinh (\frac{1}{w} + \frac{θ_{7}}{w^{7}} + \frac{θ_{9}}{w^{9}} + \frac{θ_{11}}{w^{11}} + . . .))}^{w / 2}, w \to \infty

(11)

where

θ_{7} = 1 / 810

,

θ_{9} = - 67 / 42525

, and

θ_{11} = 19 / 8505

. Chen [16] presented the following asymptotic expansion of the Gamma function related to Windschitl’s formula:

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(w sinh w^{- 1})}^{w / 2 + \sum_{k = 0}^{\infty} \frac{τ_{k}}{w^{k}}}, w \to \infty

(12)

with a recurrence relation for determining coefficients

τ_{k}

. Motivated by the above results, Yang and Tian [17] provided a more accurate Windschitl-type approximation:

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(w sinh w^{- 1})}^{w / 2} e^{\frac{7}{324} \frac{1}{w^{3} (35 w^{2} + 33)}}, w \to \infty .

(13)

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

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(w sinh w^{- 1})}^{w / 2} exp (\sum_{k = 3}^{\infty} \frac{s_{k}}{w^{2 k - 1}}), w \to \infty

(14)

where

s_{k} = \frac{2 k (2 k - 2)! - 2^{2 k - 1}}{2 k (2 k)!} B_{2 k}

and

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(w sinh w^{- 1})}^{w / 2} (1 + \sum_{k = 1}^{\infty} \frac{δ_{k}}{w^{k}}), w \to \infty

(15)

where

δ_{k} = \frac{1}{k} \sum_{r = 1}^{k} (\frac{1}{r + 1} - \frac{2^{r}}{{(r + 1)}^{2} (r - 1)!} B_{r + 1} δ_{k - r})

. Moreover, they [19] deduced the following family of high accurate approximation formulas for

q \geq - 33 / 35

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(w sinh w^{- 1})}^{w / 2} exp (\frac{1}{1620 w^{5}} \frac{w^{2} + q}{w^{2} + q + 33 / 35}), w \to \infty .

(16)

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

Γ (w + 1) = \sqrt{2 π w} {(w / e)}^{w} {(1 + \frac{1}{12 w^{2} - 1 / 10})}^{w} (1 + O (w^{- 5})), 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:

Γ (w + 1) \sim \sqrt{2 π w} e^{- w} {(w + \frac{1}{12 w + \frac{1}{10 w + \frac{φ_{1}}{w + \frac{φ_{2}}{w + \frac{φ_{3}}{w + \dots}}}}})}^{w},

(18)

where

φ_{1} = - \frac{2369}{252}

,

φ_{2} = \frac{2117009}{1193976}

, and

φ_{3} = \frac{393032191511}{1324011300744}, \dots

. 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

Γ (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 (w) \sim 1 + \sum_{p = 1}^{\infty} \frac{α_{p}}{w^{p}}, w \to \infty,

then we obtain the following asymptotic expansion

ln h (w) \sim \sum_{p = 1}^{\infty} \frac{β_{p}}{w^{p}},

where

β_{p} = α_{p} - \frac{1}{p} \sum_{r = 1}^{p - 1} r β_{r} α_{p - r}, p \in N .

Lemma 2

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

h (w) \sim \sum_{p = 1}^{\infty} \frac{α_{p}}{w^{p}}, w \to \infty,

then we obtain the following asymptotic expansion

exp h (w) \sim 1 + \sum_{p = 1}^{\infty} \frac{β_{p}}{w^{p}},

where

β_{p} = \frac{1}{p} \sum_{r = 1}^{p} r β_{p - r} α_{r}, p \in N .

Lemma 3

([26]). If

{ζ_{p}}_{p \in N}

is a null sequence,

s \in R

and

n > 1

such that

lim_{p \to \infty} p^{n} (ζ_{p} - ζ_{p + 1}) = s,

(19)

then we have

lim_{p \to \infty} p^{n - 1} ζ_{p} = \frac{s}{n - 1} .

Lemma 4

([27]). Let the real-valued function

λ (ρ)

defined for

ρ > ρ_{0}

and

{lim}_{ρ \to \infty} λ (ρ) = 0

. If

λ (ρ + a) < λ (ρ)

,

a \in N

, then

λ (ρ) > 0

for

ρ > ρ_{0}

; if

λ (ρ + a) > λ (ρ)

,

a \in N

, then

λ (ρ) < 0

for

ρ > ρ_{0}

.

3. Main Results

To obtain the best possible constants

t_{1}

and

t_{2}

in the approximation formula

n! \sim \sqrt{2 π n} {(\frac{n}{e})}^{n} {(\frac{n^{2} + t_{1}}{n^{2} + t_{2}})}^{n / 2}, n \to \infty,

(20)

we define a sequence

R_{n}

that satisfies

n! = \sqrt{2 π n} {(\frac{n}{e})}^{n} {(\frac{n^{2} + t_{1}}{n^{2} + t_{2}})}^{n / 2} e^{R_{n}}, n \geq 1 .

Then,

\begin{matrix} R_{n} - R_{n + 1} = & \frac{- 6 t_{1} + 6 t_{2} + 1}{12 n^{2}} + \frac{6 t_{1} - 6 t_{2} - 1}{12 n^{3}} + \frac{30 t_{1}^{2} - 20 t_{1} - 30 t_{2}^{2} + 20 t_{2} + 3}{40 n^{4}} \\ + \frac{- 45 t_{1}^{2} + 15 t_{1} + 45 t_{2}^{2} - 15 t_{2} - 2}{30 n^{5}} \\ + \frac{- 70 t_{1}^{3} + 210 t_{1}^{2} - 42 t_{1} + 70 t_{2}^{3} - 210 t_{2}^{2} + 42 t_{2} + 5}{84 n^{6}} + O (n^{- 13 / 2}) . \end{matrix}

If

- 6 t_{1} + 6 t_{2} + 1 \neq 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

30 t_{1}^{2} - 20 t_{1} - 30 t_{2}^{2} + 20 t_{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 (\frac{n! {(\frac{n^{2} + \frac{7}{60}}{n^{2} - \frac{1}{20}})}^{- n / 2}}{\sqrt{2 π n} {(\frac{n}{e})}^{n}}) = O (n^{- 5}), n \to \infty

(21)

where

lim_{n \to \infty} \frac{R_{n} - R_{n + 1}}{n^{- 6}} = \frac{461}{181440} .

Theorem 1.

The Gamma function has the following asymptotic expansion:

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{w / 2} exp (\sum_{r = 1}^{\infty} \frac{μ_{r}}{w^{r}}), w \to \infty

(22)

where

μ_{r}

are given by

μ_{r} = \frac{B_{r + 1}}{r (r + 1)} - \frac{1}{2} ν_{r + 1}, r = 1, 2, 3, . . .

(23)

where

B_{r}

denote Bernoulli numbers and

ν_{r} = σ_{r} - \frac{1}{r} \sum_{j = 1}^{r - 1} j ν_{j} σ_{r - j}, r = 1, 2, 3, . . .

with

σ_{0} = 1, σ_{2 s} = \frac{10}{3 {(20)}^{s}}, σ_{2 s - 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_{2 r + 1} = 0

,

r = 1, 2, 3, . . .

, we have the following.

ln (\frac{Γ (w + 1)}{\sqrt{2 π w} {(\frac{w}{e})}^{w}}) \sim \sum_{r = 1}^{\infty} \frac{B_{r + 1}}{r (r + 1)} w^{- r}, w \to \infty .

(24)

Moreover, we obtain

\begin{matrix} \frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}} & = & 1 + \sum_{n = 1}^{\infty} \frac{10}{3 {(20)}^{n}} {(w^{2})}^{- n} \\ = & \sum_{s = 0}^{\infty} σ_{s} w^{- s}, | w | > \sqrt{\frac{1}{20}} \end{matrix}

with

σ_{0} = 1, σ_{2 s} = \frac{10}{3 {(20)}^{s}}, σ_{2 s - 1} = 0, s = 1, 2, 3, . . . .

By Lemma 1, we obtain

ln (\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}}) \sim \sum_{r = 1}^{\infty} ν_{r} w^{- r}, w \to \infty

(25)

with

ν_{r} = σ_{r} - \sum_{j = 1}^{r - 1} \frac{j ν_{j} σ_{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{ν_{r}}{2 w^{r - 1}} \sim \sum_{r = 1}^{\infty} \frac{μ_{r}}{w^{r}}, w \to \infty

which gives us with the aid of

v_{1} = 0

such that

\sum_{r = 1}^{\infty} (\frac{B_{r + 1}}{r (r + 1)} - \frac{ν_{r + 1}}{2}) w^{- r} \sim \sum_{r = 1}^{\infty} μ_{r} w^{- r}, w \to \infty .

(26)

Equating coefficients of

w^{- r}

yields

μ_{r} = \frac{B_{r + 1}}{r (r + 1)} - \frac{ν_{r + 1}}{2}, r = 1, 2, 3, . . . .

□

Remark 1.

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

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

(27)

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

Theorem 2.

The Gamma function has the asymptotic expansion

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{w / 2} (\sum_{r = 0}^{\infty} \frac{λ_{r}}{w^{r}}), w \to \infty

(28)

where

λ_{r}

are given by

λ_{0} = 1, λ_{r} = \frac{1}{r} \sum_{j = 1}^{r} j λ_{r - j} μ_{j}, r = 1, 2, 3, . . .

(29)

with coefficients

μ_{r}

being given by (23).

Remark 2.

According to Theorem 2, we have

Γ (w + 1) \sim \sqrt{2 π w} {(w / e)}^{w} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{w / 2} (1 + O (w^{- 5})), 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 $n!$

We have the following approximation formulas:

$n! \sim$	$\sqrt{2 π n} {(n / e)}^{n} {(n sinh (\frac{1}{n} + \frac{1}{810 n^{7}}))}^{n / 2}$	$≑ a_{n},$	[15]
$n! \sim$	$\sqrt{2 π n} {(n / e)}^{n} {(1 + \frac{1}{12 - \frac{1}{10 n^{2} - \frac{2369}{252}}})}^{n}$	$≑ b_{n},$	[22]
$n! \sim$	$\sqrt{2 π} {(n + 1)}^{n + 1 / 2} e^{- n - 1} exp (\frac{1}{12 n} - \frac{1}{12 n^{2}} + \frac{29}{360 n^{3}} - \frac{3}{40 n})$	$≑ c_{n},$	[28]
$n! \sim$	$\sqrt{2 π} {(n + 1)}^{n} e^{- n - 1 / 2} exp {(n^{3} + 5 / 4 n^{2} + 17 / 32 n + \frac{172}{1920})}^{1 / 6}$	$≑ d_{n},$	[29]
$n! \sim$	$\sqrt{2 π n} {(n / e)}^{n} {(\frac{n^{2} - 9 / 120}{n^{2} - 1 / 120})}^{n}$	$≑ f_{n},$	[21]

and

n! \sim

\sqrt{2 π n} {(n / e)}^{n} {(n sinh \frac{1}{n})}^{n / 2} exp (\frac{n}{2} + \frac{1}{270 n^{3}})

≑ 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 π n} {(n / e)}^{n} {(n sinh n^{- 1})}^{n / 2} exp (\frac{1}{1620} - \frac{11}{18900 n^{7}}) ≑ h_{n}

and

n! \sim \sqrt{2 π n} {(n / e)}^{n} {(n sinh n^{- 1})}^{n / 2} (1 + \frac{1}{1620} - \frac{11}{18900 n^{7}}) ≑ k_{n}

are stronger than formula

g_{n}

for

n \geq 2

. Table 1 shows that the approximation

n! \sim \sqrt{2 π n} {(n / e)}^{n} {(\frac{n^{2} + \frac{7}{60}}{n^{2} - \frac{1}{20}})}^{n / 2} exp (\frac{461}{907200 n^{5}} - \frac{5197}{9072000 n^{7}}) ≑ l_{n}

is stronger than approximation

h_{n}

.

Moreover, Table 2 shows that approximation

n! \sim \sqrt{2 π n} {(n / e)}^{n} {(\frac{n^{2} + \frac{7}{60}}{n^{2} - \frac{1}{20}})}^{n / 2} (1 + \frac{461}{907200 n^{5}} - \frac{5197}{9072000 n^{7}}) ≑ p_{n}

is stronger than approximation

k_{n}

:

5. Some Bounds of $Γ (w + 1)$ Using Padé Approximants

Consider the following formal power series.

K (w) = α_{0} + α_{1} w + α_{2} w^{2} + . . .,

Then, the rational function

{[m, s]}_{K} (w) = \frac{\sum_{i = 0}^{m} γ_{i} w^{i}}{1 + \sum_{i = 1}^{s} β_{i} w^{i}}, m \geq 0; s \geq 1

(31)

is called the Padé approximant of order

(m, s)

of function

K (w)

[31,32,33], where

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

(32)

and coefficients

β_{i}

are the solutions of the system

[\begin{matrix} 0 \\ 0 \\ ⋮ \\ 0 \end{matrix}] = [\begin{matrix} α_{m + 1} & α_{m} & . . . & α_{m - s + 1} \\ α_{m + 2} & α_{m + 1} & . . . & α_{m - s + 2} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ α_{m + s} & α_{m + s - 1} & . . . & α_{m} \end{matrix}] [\begin{matrix} 1 \\ β_{1} \\ ⋮ \\ β_{s} \end{matrix}]

(33)

with

α_{j} = 0

for

j < 0

, and coefficients

γ_{i}

are given by the following.

\begin{matrix} γ_{0} & = & α_{0} \\ γ_{1} & = & α_{1} + α_{0} β_{1} \\ ⋮ \\ γ_{m} & = & α_{m} + α_{m - 1} β_{1} + \dots + α_{m - s} β_{s} \end{matrix}\} .

(34)

For the formal power series

f (w) ≑ \sum_{r = 1}^{\infty} \frac{μ_{r}}{w^{r}}

, where

μ_{r}

are given by (23), we can conclude the following Padé approximations

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

and

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

Hence, we obtain the following double inequality.

Theorem 3.

The following double inequality holds for

w \geq 1

.

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

< \frac{Γ (w + 1)}{\sqrt{2 π w} {(w / e)}^{w}} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{- w / 2} <

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

(35)

Proof.

Consider the function

M (w) = \frac{exp (- \frac{461}{907200 (\frac{89617471668379}{60523294534560 w^{6}} - \frac{213922547}{561055440 w^{4}} + \frac{5197}{4610 w^{2}} + 1) w^{5}}) Γ (w + 1)}{\sqrt{2 π w} {(\frac{w}{e})}^{w} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{w / 2}}, w \geq 1

and then

M^{'} (w) = \frac{e^{\frac{w (12709891852257600 w^{6} + 14328266367935520 w^{4} - 4846102975366380 w^{2} + 18813210430271527)}{210 (60523294534560 w^{6} + 68229839847312 w^{4} - 23076680835078 w^{2} + 89617471668379)}} Γ (w + 1)}{\sqrt{2 π} w^{w + \frac{1}{2}} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{\frac{w}{2}}} M_{1} (w),

where

\begin{matrix} M_{1} (w) & = & \frac{1}{20 w^{2} - 1} + \frac{7}{60 w^{2} + 7} - \frac{1}{2 w} - \frac{1}{2} log (\frac{10}{60 w^{2} - 3} + 1) - log (w) + ψ (w + 1) \\ - & \frac{6458620088063 (71916 (316247844 w^{2} - 213922547) w^{2} + 89617471668379)}{35 {(107874 (121704 w^{2} (4610 w^{2} + 5197) - 213922547) w^{2} + 89617471668379)}^{2}} \\ + & \frac{6458620088063}{42 (107874 (121704 w^{2} (4610 w^{2} + 5197) - 213922547) w^{2} + 89617471668379)} . \end{matrix}

Using

M_{2} (w) = M_{1} (w + 1) - M_{1} (w)

, we obtain

M_{2}^{'} (w + 1) = \frac{M_{3} (w)}{M_{4} (w)} < 0, w \geq 0

where

\begin{matrix} M_{3} (w) & = & - 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} \\ - 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} \\ - 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} \\ - 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} \\ - 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} \\ - 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} \\ - 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} \\ - 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 \\ - 6.7 \times 10^{104} \end{matrix}

and

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

Now,

M_{2} (w)

is a decreasing function for

w \geq 1

with

{lim}_{w \to \infty} M_{2} (w) = 0

; then,

M_{2} (w) > 0

or

M_{1} (w) < M_{1} (w + 1)

for

w \geq 1

. However,

{lim}_{w \to \infty} M_{1} (w) = 0

; hence, by using Lemma 4, we have

M_{1} (w) < 0

for

w \geq 1

. Then,

M (w)

is a decreasing function for

w \geq 1

with

{lim}_{w \to \infty} M (w) = 1

and then

M (w) > 1

for

w \geq 1

or

\frac{Γ (w + 1)}{\sqrt{2 π w} {(\frac{w}{e})}^{w} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{w / 2}} > exp (\frac{461}{907200 (\frac{89617471668379}{60523294534560 w^{6}} - \frac{213922547}{561055440 w^{4}} + \frac{5197}{4610 w^{2}} + 1) w^{5}}), w \geq 1 .

Now, consider the following function

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

and then

H^{'} (w) = \frac{e^{w - \frac{461}{907200 (- \frac{213922547}{561055440 w^{4}} + \frac{5197}{4610 w^{2}} + 1) w^{5}}} w^{- w - \frac{1}{2}} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{- \frac{w}{2}} Γ (w + 1)}{\sqrt{2 π}} H_{1} (w),

where

\begin{matrix} H_{1} (w) & = & \frac{200 w^{2}}{1200 w^{4} + 80 w^{2} - 7} - \frac{1}{2 w} - \frac{1}{2} log (\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}}) - log (w) + ψ (w + 1) \\ - \frac{1077693991 (316247844 w^{2} - 213922547)}{945 w^{2} {(561055440 w^{4} + 632495688 w^{2} - 213922547)}^{2}} \\ + \frac{1077693991}{756 w^{2} (561055440 w^{4} + 632495688 w^{2} - 213922547)} . \end{matrix}

Using

H_{2} (w) = H_{1} (w + 1) - H_{1} (w)

, we obtain

H_{2}^{'} (w + 1) = \frac{H_{3} (w)}{H_{4} (w)} < 0, w \geq 0

where

\begin{matrix} H_{3} (w) & = & 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} \\ + 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} \\ + 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} \\ + 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} \\ + 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} \\ + 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} \\ + 1.1 \times 10^{76} w^{2} + 1.0 \times 10^{75} w + 4.6 \times 10^{73} \end{matrix}

and

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

Now,

H_{2} (w)

is an increasing function for

w \geq 1

with

{lim}_{w \to \infty} H_{2} (w) = 0

; then,

H_{2} (w) < 0

or

H_{1} (w + 1) < H_{1} (w)

for

w \geq 1

. However,

{lim}_{w \to \infty} H_{1} (w) = 0

; hence, using Lemma 4, we have

H_{1} (w) > 0

for

w \geq 1

. Then,

H (w)

is an increasing function for

w \geq 1

with

{lim}_{w \to \infty} H (w) = 1

and then

H (w) < 1

for

w \geq 1

or

\frac{Γ (w + 1)}{\sqrt{2 π w} {(\frac{w}{e})}^{w} {(\frac{w^{2} + \frac{7}{60}}{w^{2} - \frac{1}{20}})}^{w / 2}} < e^{\frac{461}{907200 (\frac{213922547}{561055440 w^{4}} + \frac{5197}{4610 w^{2}} + 1) w^{5}}}, w \geq 1 .

□

Remark 3.

The bounds in our new inequality (35) are better than the bounds in Alzer’s inequality (10) for

w \geq 1

.

6. Conclusions

The Padé approximant method and asymptotic expansions can be employed to present some new bounds of

Γ (w)

. 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.

Author Contributions

Writing to original draft, M.M. and H.A. All authors contributed equally to the writing of this paper. All authors have read and agreed to the published version of the manuscript.

Funding

The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia has funded this Project under grant no (G: 716-130-1443).

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Correction Statement

This article has been republished with a minor correction to the existing affiliation information. This change does not affect the scientific content of the article.

References

Batir, N. Very accurate approximations for the factorial function. J. Math. Inequal. 2010, 4, 335–344. [Google Scholar] [CrossRef]
Gosper, R.W. Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 1978, 75, 40–42. [Google Scholar] [CrossRef] [PubMed]
Mortici, C. On Gospers formula for the Gamma function. J. Math. Inequal. 2011, 5, 611–614. [Google Scholar] [CrossRef]
Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables; Applied Mathematical Series; Nation Bureau of Standards: Dover, DE, USA; New York, NY, USA, 1972; Volume 55. [Google Scholar]
Burnside, W. A rapidly convergent series for logN! Messenger Math. 1917, 46, 157–159. [Google Scholar]
Andrews, G.E.; Berndt, B.C. Ramanujan’s Lost Notebook: Part IV; Springer Science & Business Media: New York, NY, USA, 2013. [Google Scholar]
Berndt, B.C.; Choi, Y.-S.; Kang, S.-Y. The problems submitted by Ramanujan. J. Indian Math. Soc. Contemp. Math. 1999, 236, 15–56. [Google Scholar]
Karatsuba, E.A. On the asymptotic representation of the Euler Gamma function by Ramanujan. J. Comput. Appl. Math. 2001, 135, 225–240. [Google Scholar] [CrossRef]
Mortici, C. On Ramanujan’s large argument formula for the Gamma function. Ramanujan J. 2011, 26, 185–192. [Google Scholar] [CrossRef]
Ramanujan, S. The Lost Notebook and Other Unpublished Papers; Narosa Publishing House: New Delhi, India; Springer: Berlin/Heidelberg, Germany, 1988. [Google Scholar]
Mortici, C. Improved asymptotic formulas for the Gamma function. Comput. Math. Appl. 2011, 61, 3364–3369. [Google Scholar] [CrossRef]
Available online: http://www.rskey.org/gamma.htm (accessed on 20 April 2020).
Smith, W.D. The Gamma Function Revisited. 2006. Available online: http://schule.bayernport.com/gamma/gamma05.pdf (accessed on 20 April 2020).
Alzer, H. Sharp upper and lower bounds for the Gamma function. Proc. R. Soc. Edinb. 2009, 139A, 709–718. [Google Scholar] [CrossRef]
Lu, D.; Song, L.; Ma, C. A generated approximation of the Gamma function related to Windschitl’s formula. J. Number Theory 2014, 140, 215–225. [Google Scholar] [CrossRef]
Chen, C.-P. Asymptotic expansions of the Gamma function related to Windschitl’s formula. Appl. Math. Comput. 2014, 245, 174–180. [Google Scholar] [CrossRef]
Yang, Z.-H.; Tian, J.-F. An accurate approximation formula for Gamma function. J. Inequal. Appl. 2018, 2018, 56. [Google Scholar] [CrossRef] [PubMed]
Yang, Z.-H.; Tian, J.-F. Two asymptotic expansions for Gamma function developed by Windschitl’s formula. Open Math. 2018, 16, 1048–1060. [Google Scholar] [CrossRef]
Yang, Z.-H.; Tian, J.-F. A family of Windschitl type approximations for Gamma function. J. Math. Inequal. 2018, 12, 889–899. [Google Scholar] [CrossRef]
Yang, Z.-H.; Tian, J.-F. Windschitl type approximation formulas for the Gamma function. J. Inequal. Appl. 2018, 2018, 272. [Google Scholar] [CrossRef] [PubMed]
Nemes, G. New asymptotic expansion for the Gamma function. Arch. Math. 2010, 95, 161–169. [Google Scholar] [CrossRef]
Mortici, C. A continued fraction approximation of the Gamma function. J. Math. Anal. Appl. 2013, 402, 405–410. [Google Scholar] [CrossRef]
Ahmad, S.; Ullah, A.; Akgül, A.; Jarad, F. A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations. AIMS Math. 2022, 7, 9389–9404. [Google Scholar] [CrossRef]
Gulalai Ahmad, S.; Rihan, F.A.; Ullah, A.; Al-Mdallal, Q.M.; Akgül, A. Nonlinear analysis of a nonlinear modified KdV equation under Atangana Baleanu Caputo derivative. AIMS Math. 2022, 7, 7847–7865. [Google Scholar] [CrossRef]
Chen, C.-P.; Elezović, N.; Vukšić, L. Asymptotic formulae associated with the Wallis power function and digamma function. J. Class. Anal. 2013, 2, 151–166. [Google Scholar] [CrossRef]
Mortici, C. New approximations of the Gamma function in terms of the digamma function. Appl. Math. Lett. 2010, 23, 97–100. [Google Scholar] [CrossRef]
Elbert, A.; Laforgia, A. On some properties of the Gamma function. Proc. Amer. Math. Soc. 2000, 128, 2667–2673. [Google Scholar] [CrossRef]
Mortici, C. New improvements of the Stirling formula. Appl. Math. Comput. 2010, 217, 699–704. [Google Scholar] [CrossRef]
Mortici, C. Ramanujan formula for the generalized Stirling approximation. Ramanujan J. 2010, 217, 2579–2585. [Google Scholar] [CrossRef]
Chen, C.-P. Asymptotic Expansions of the Gamma Function Associated with the Windschitl and Smith Formulas. 2014. Available online: https://rgmia.org/papers/v17/v17a109.pdf (accessed on 18 April 2020).
Baker, G.A., Jr.; Graves-Morris, P.R. Padé Approximants, 2nd ed.; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
Brezinski, C. Computational Aspects of Linear Control; Kluwer: Dordrecht, The Netherlands, 2002. [Google Scholar]
Brezinski, C.; Redivo-Zaglia, M. New representations of Padé, Padé-type, and partial Padé approximants. J. Comput. Appl. Math. 2015, 284, 69–77. [Google Scholar] [CrossRef]

Table 1. Comparison among

h_{n}

and

l_{n}

.

Table 1. Comparison among

h_{n}

and

l_{n}

.

n	$\|\frac{n! - h_{n}}{n!}\|$	$\|\frac{n! - l_{n}}{n!}\|$
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	$8.40680 \times 10^{- 31}$	$8.39556 \times 10^{- 31}$

Table 2. Comparison among

k_{n}

and

p_{n}

.

Table 2. Comparison among

k_{n}

and

p_{n}

.

n	$\|\frac{n! - k_{n}}{n!}\|$	$\|\frac{n! - p_{n}}{n!}\|$
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	$8.40680 \times 10^{- 31}$	$8.39556 \times 10^{- 31}$

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Mahmoud, M., & Almuashi, H. (2022). On Some Asymptotic Expansions for the Gamma Function. Symmetry, 14(11), 2459. https://doi.org/10.3390/sym14112459

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On Some Asymptotic Expansions for the Gamma Function

Abstract

1. Introduction

2. Lemmas

3. Main Results

4. Numerical Comparisons among Some Approximation Formulas of $n!$

5. Some Bounds of $Γ (w + 1)$ Using Padé Approximants

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Correction Statement

References

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Article Menu

On Some Asymptotic Expansions for the Gamma Function

Abstract

1. Introduction

2. Lemmas

3. Main Results

4. Numerical Comparisons among Some Approximation Formulas of n !

5. Some Bounds of Γ ( w + 1 ) Using Padé Approximants

6. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Correction Statement

References

Share and Cite

Article Metrics

Article Access Statistics

Further Information

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4. Numerical Comparisons among Some Approximation Formulas of $n!$

5. Some Bounds of $Γ (w + 1)$ Using Padé Approximants