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# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 16, Issue 1

# Two asymptotic expansions for gamma function developed by Windschitl’s formula

Zhen-Hang Yang
• College of Science and Technology, North China Electric Power University, Baoding, Hebei Province, 071051, China
• Department of Science and Technology, State Grid Zhejiang Electric Power Company Research Institute, Hangzhou, Zhejiang, 310014, China
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/ Jing-Feng Tian
• Corresponding author
• College of Science and Technology, North China Electric Power University, Baoding, Hebei Province, 071051, China
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Published Online: 2018-08-24 | DOI: https://doi.org/10.1515/math-2018-0088

## Abstract

In this paper we develop Windschitl’s approximation formula for the gamma function by giving two asymptotic expansions using a little known power series. In particular, for n ∈ ℕ with n ≥ 4, we have

$Γx+1=2πxxexxsinh⁡1xx/2exp∑k=3n−1anx2n−1+Rnx$

with

$Rnx≤B2n2n2n−11x2n−1$

for all x > 0, where an has a closed-form expression, B2n is the Bernoulli number. Moreover, we present some approximation formulas for the gamma function related to Windschitl’s approximation, which have higher accuracy.

MSC 2010: 33B15; 41A60; 41A10; 41A20

## 1 Introduction

It is known that the Stirling’s formula

$n!∼2πnnen$(1)

for n ∈ ℕ has various applications in probability theory, statistical physics, number theory, combinatorics and other branches of science. As a generalization of the factorial function, the gamma function Γ (x) = $\begin{array}{}{\int }_{0}^{\mathrm{\infty }}{t}^{x-1}{e}^{-t}dt\end{array}$ for x > 0 is no exception. Thus, many scholars pay attention to find various better approximations for the factorial or gamma function, for example, Ramanujan [1, P. 339], Burnside [2], Gosper [3], Alzer [4, 5], Windschitl (see [6, 7]), Smith [6], Batir [8, 9], Mortici [10, 11, 12, 13, 14, 15] Nemes [16, 17], Qi et al. [18, 19], Chen [20], Yang et al. [21, 22, 23], Lu et al. [24, 25].

As an asymptotic expansion of Stirling’s formula (1), one has the Stirling’s series for the gamma function [26, p. 257, Eq. (6.1.40)]

$Γx+1∼2πxxexexp∑n=1∞B2n2n2n−1x2n−1$(2)

as x → ∞, where B2n for n ∈ ℕ∪{0} is the Bernoulli number. It was proved in [4, Theorem 8] by Alzer (see also [27, Theorem 2]) that for given integer n ∈ ℕ, the function

$Fnx=lnΓx+1−x+12ln⁡x+x−12ln2π−∑k=1nB2k2k2k−1x2k−1$

is strictly completely monotonic on (0, ∞) if n is even, and so is − Fn(x) if n is odd. It follows that the double inequality

$exp∑k=12nB2k2k2k−1x2k−1<Γx+12πxx/ex(3)

holds for all x > 0.

Another asymptotic expansion is the Laplace series (see [26, p. 257, Eq. (6.1.37)])

$Γx+1∼2πxxex1+112x+1288x2−13951840x3−5712488320x4+⋅⋅⋅$(4)

as x → ∞. Other asymptotic expansions developed by some closed approximation formulas for the gamma function can be found in [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] and the references cited therein.

Now let us focus on the Windschitl’s approximation formula given by

$Γx+1∼W0x=2πxxexxsinh⁡1xx/2, as x→∞.$(5)

As shown in [20, Eq. (3.8)], the rate of Windschitl’s approximation W0(x) converging to Γ (x + 1) is like x−5 as x → ∞, and is like x−7 if replacing W0(x) with

$W1x=2πxxexxsinh⁡1x+1810x6x/2,$(6)

by an easy check. These show that W0(x) and W1(x) are excellent approximations for the gamma function. Recently, Lu, Song and Ma [32] extended Windschitl’s formula to the following asymptotic expansion

$Γn+1∼2πnnennsinh1n+a7n7+a9n9+a11n11+⋅⋅⋅n/2$(7)

as n → ∞ with a7 = 1/810, a9 = −67/42525, a11 = 19/8505, …. An explicit formula for determining the coefficients of nk (n ∈ ℕ) was given in [34, Theorem 1] by Chen. Other two asymptotic expansions

$Γx+1∼2πxxexxsinh⁡1xx/2+∑j=0∞rjx−j,$(8)

$Γx+1∼2πxxexxsinh⁡1x+∑n=3∞dnx2nx/2,$(9)

as x → ∞ were presented in the papers [34, Theorem 2], [36], respectively.

Inspired by the asymptotic expansions (7), (8), (9) and Windschitl’s approximation formula (6), the first aim of this paper is to further present the following two asymptotic expansions related to Windschitl’s one (5): as x → ∞,

$Γx+1∼2πxxexxsinh⁡1xx/2exp∑n=3∞anx2n−1,$(10)

$Γx+1∼2πxxexxsinh⁡1xx/21+∑n=1∞bnxn.$(11)

It is worth pointing out that those coefficients in (10) have a closed-form expression, which is due to a little known power series expansion of ln (t−1sinh t) (Lemma 2.1). We also give an estimate of the remainder in the asymptotic expansion (10). Incidentally, we provide a more explicit coefficients formula in Chen’s asymptotic expansion (8). These results (Theorems 1–4) are presented in Section 2.

The second aim of this paper is to give some closed approximation formulas for the gamma function generated by truncating five asymptotic series just mentioned, and compare the accuracy of them by numeric computations and some inequalities. These results (Table 1 and Theorem 5) are listed in Section 3.

Table 1

Comparisons among W1(x), Wc1(x), W01(x), Wl1(x)

## 2 Asymptotic expansions

To obtain the explicit coefficients formulas in the asymptotic expansions (10), (11) and (8), and to estimate the remainder in the asymptotic expansions (10), we first give a lemma.

#### Lemma 2.1

For |t| < π, we have

$ln⁡sinh⁡tt=∑n=1∞22nB2n2n2n!t2n.$(12)

Moreover, for n ∈ ℕ, the double inequality

$∑k=12n22kB2k2k2k!t2k(13)

holds for all t > 0.

#### Proof

It was listed in [26, p. 85, Eq. (4.5.64), (4.5.65), (4.5.67)] that

$coth⁡t=∑n=0∞22nB2n2n!t2n−1t<π.$

Then we obtain that for |t| < π,

$ln⁡sinh⁡tt=∫0tcoth⁡x−1xdx=∫0t∑n=0∞22nB2n2n!x2n−1−1xdx=∑n=1∞22nB2n2n2n!t2n.$

The double inequality (13) was proved in [23, Corollary 1], and the proof is completed. □

#### Theorem 2.2

As x → ∞, the asymptotic expansion

$Γx+1∼2πxxexxsinh⁡1xx/2exp∑n=3∞anx2n−1=2πxxexxsinh⁡1xx/2exp11620x5−1118900x7+143170100x9−22602611178793000x11+⋅⋅⋅$

holds with

$an=2n2n−2!−22n−12n2n!B2n,$(14)

where B2n is the Bernoulli number.

#### Proof

By the asymptotic expansion (2) and Lemma 2.1 we have that as x → ∞,

$lnΓx+1−ln⁡2π−x+12ln⁡x+x∼∑n=1∞an′x2n−1, x2lnxsinh⁡1x=12∑n=1∞an′′x2n−1,$

where

$an′=B2n2n2n−1 and an′′=22nB2n2n2n!.$

Let

$Γx+1∼2πxxexxsinh⁡1xx/2expw0x as x→∞.$

Then we have that as x → ∞,

$w0x=lnΓx+1−ln⁡2π−x+12ln⁡x+x−x2lnxsinh⁡1x=∑n=1∞an′x2n−1−12∑n=1∞an′′x2n−1=∑n=1∞an′−an′′/2x2n−1=∑n=1∞anx2n−1.$

An easy computation yields a1 = a2 = 0 and

$a3=11620, a4=−1118900, a5=143170100, a6=−22602611178793000$

which completes the proof. □

The following theorem offers an estimate of the remainder in the asymptotic expansion (10).

#### Theorem 2.3

For n ∈ ℕ with n ≥ 4, let

$Γx+1=2πxxexxsinh⁡1xx/2exp∑k=3n−1akx2k−1+Rnx,$

where ak is given by (14). Then we have

$Rnx≤B2n2n2n−11x2n−1$

for all x > 0.

#### Proof

We have

$Rnx=lnΓx+1−ln⁡2π−x+12ln⁡x+x−x2lnxsinh⁡1x−∑k=3n−1akx2k−1.$

If n = 2m + 1 for m ≥ 2 then by inequalities (3) and (13) we have

$R2mx=lnΓx+1−ln⁡2π−x+12ln⁡x+x−x2lnxsinh⁡1x−∑k=32makx2k−1 >∑k=12mak′x2k−1−12∑k=12m+1ak′′x2k−1−∑k=32mamx2k−1=−12a2m+1′′x4m+1,R2mx<∑k=12m+1ak′x2k−1−12∑k=12ma¨kx2k−1−∑k=32mamx2k−1=a2m+1′x4m+1,$

where the last equalities in the above two inequalities hold due to $\begin{array}{}{a}_{m}^{\mathrm{\prime }}-{a}_{m}^{\mathrm{\prime }\mathrm{\prime }}/2={a}_{m}\end{array}$. It follows that

$Rnx

The calculations also hold for the case when n = 2m, m ≥ 2.

Since

$an′an′′/2=B2n2n2n−11222nB2n2n2n!=22n!2n−122n:=an′′′,an+1′′′an′′′−1=122n+3n−1>0,$

it is derived that $\begin{array}{}{a}_{n}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}>{a}_{4}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\end{array}$ = 45 for n ≥ 4, so we obtain

$max12an′′,an′=an′=B2n2n2n−1,$

which completes the proof. □

#### Remark 2.4

Since B2n+1 = 0 for n ∈ ℕ, the asymptotic series w0(x) can also be written as

$w0x=∑n=3∞2n2n−2!−22n−12n2n!B2nx2n−1=∑n=1∞n+1n−1!−2nn+1n+1!Bn+1xn:=∑n=1∞an∗xn,$

where

$an∗=n+1n−1!−2nn+1n+1!Bn+1.$(15)

Now we establish the second Windschitl type asymptotic series for the gamma function.

#### Theorem 2.5

As x → ∞, the asymptotic expansion (9)

$Γx+1∼2πxxexxsinh⁡1xx/21+∑n=1∞bnxn=2πxxexxsinh⁡1xx/21+11620x5−1118900x7+143170100x9+15248800x10+⋅⋅⋅$

holds with b0 = 1, b1 = b2 = b3 = b4 = 0 and for n ≥ 5,

$bn=1n∑k=1n1k+1−2kk+12k−1!Bk+1bn−k.$(16)

#### Proof

It was proved in [29, Lemma 3] that as x → ∞,

$exp∑n=1∞anx−n∼∑n=0∞bnx−n$

with b0 = 1 and

$bn=1n∑k=1nkakbn−k for n≥1.$(17)

Substituting $\begin{array}{}{a}_{k}^{\ast }\end{array}$ given in (15) into (17) gives recurrence formula (16).

An easy verification shows that bn = 0 for 1 ≤ n ≤ 4, b6 = b8 = 0 and

$b5=11620, b7=−1118900, b9=143170100, b10=15248800,$

which completes the proof. □

The following theorem improves Chen’s result [34, Theorem 2].

#### Theorem 2.6

As x → ∞, the asymptotic expansion

$Γx+1∼2πxxexxsinh⁡1xx/21+∑n=2∞cnx−2n=2πxxexxsinh⁡1xx21+1135x4−19128350x6+251272551500x8−19084273841995000x10+⋅⋅⋅$(18)

holds with c0 = 1, c1 = 0 and for n ≥ 2,

$cn=6B2n+2n+12n+1−6∑k=1n22k+2B2k+22k+12k+2!cn−k.$

#### Proof

The asymptotic expansion (8) can be written as

$lnΓx+1−ln⁡2πx−xln⁡x+x∼x2+∑j=0∞rjxjlnxsinh⁡1x,$

which, by (2) and (12), is equivalent to

$∑n=1∞B2n2n2n−11x2n−1∼x2+∑j=0∞rjxj∑n=1∞22nB2n2n2n!1x2n.$

Since the left hand side and the second factor of the right hand side are odd and even, respectively, the asymptotic expansion $\begin{array}{c}x/2+\sum _{j=0}^{\mathrm{\infty }}{r}_{j}{x}^{-j}\end{array}$ has to be odd, and so r2n = 0 for n ∈ ℕ∪{0}. Then, the asymptotic expansion (8) has the form of (18), which is equivalent to

$∑n=1∞B2n2n2n−11x2n−1∼x2∑n=0∞cnx2n∑n=1∞22nB2n2n2n!1x2n.$

It can be written as

$∑n=0∞B2n+2n+12n+11x2n∼∑n=0∞cnx2n∑n=1∞22n+2B2n+22n+12n+2!1x2n.$

Comprising coefficients of x−2n gives

$B2n+2n+12n+1=∑k=0n22k+2B2k+22k+12k+2!cn−k,$

which yields c0 = 1 and for n ≥ 1,

$cn=6B2n+2n+12n+1−6∑k=1n22k+2B2k+22k+12k+2!cn−k.$

$c1=0, c2=1135, c3=−19128350 , c4=251272551500, c5=−19084273841995000,$

which ends the proof. □

#### Remark 2.7

Chen’s recurrence formula of coefficients rj given in [34, Theorem 2] may be complicated, since he was unaware of the power series (12).

## 3 Numeric comparisons and inequalities

If the series in (10), (11), (9) (18) are truncated at n = 3, 5, 3, 2, respectively, then we obtain four Windschitl type approximation formulas:

$Γx+1∼2πxxexxsinh⁡1xx/2exp11620x5:=W01x,$(19)

$Γx+1∼2πxxexxsinh⁡1xx/21+11620x5:=W01∗x,$(20)

$Γx+1∼2πxxexxsinh⁡1x+1810x6x/2=W1x,Γx+1∼2πxxexxsinh⁡1xx21+1135x4:=Wc1x,$(21)

as x → ∞. Also, we denote Lu et al.’s one [32, Theorem 1.8] by

$Wl1x=2πxxexxsinh1x+1810x7x/2.$(22)

In this section, we aim to compare the five closed approximation formulas listed above.

## 3.1 Numeric comparisons

We easily obtain

$limx→∞lnΓx+1−ln⁡W1xx−7=−163340200,limx→∞lnΓx+1−ln⁡Wc1xx−7=−191340200,$

$limx→∞lnΓx+1−ln⁡W01xx−7=limx→∞lnΓx+1−ln⁡W01∗xx−7=−198340200,limx→∞lnΓx+1−ln⁡Wc1xx−7=−268340200.$

These show that the rates of the five approximation formulas converging to Γ (x + 1) are all like x−7 as x → ∞, and W1(x) are the best of all five approximations, which can also be seen from the following Table 1.

## 3.2 Three lemmas

As is well known, analytic inequality [40, 41, 42] is playing a very important role in different branches of modern mathematics. To further compare W1(x), Wc1(x), $\begin{array}{}{W}_{01}^{\ast }\end{array}$ (x), W01(x) and Wl1(x), we first give the following inequality.

#### Lemma 3.1

The inequality

$ψ′x+12<1xx4+6736x2+256945x4+3518x2+4071008$(23)

holds for all x > 0.

#### Proof

Let

$gx=ψx+12,1−1xx4+6736x2+256945x4+3518x2+4071008.$

Then we have

$gx+1−gx=ψx+32,1−1x+1x+14+6736x+12+256945x+14+3518x+12+4071008−ψx+12,1+1xx4+6736x2+256945x4+3518x2+4071008=921600xx+12x+121008x4+1960x2+4071008x4+4032x3+8008x2+7952x+3375>0.$

Hence, we conclude that

$gx

which proves (23), and the proof is done. □

The second lemma offers a simple criterion to determine the sign of a class of special polynomial on a given interval contained in (0, ∞) without using Descartes’ Rule of Signs, which plays an important role in the study of certain special functions, see for example [43, 44]. A series version can be found in [45].

#### Lemma 3.2

[43, [Lemma 7]. Let n ∈ ℕ and m ∈ ℕ ∪ {0} with n > m and let Pn(t) be an n degrees polynomial defined by

$Pnt=∑i=m+1naiti−∑i=0maiti,$(24)

where an, am > 0, ai ≥ 0 for 0 ≤ in − 1 with im. Then, there is a unique number tm+1 ∈ (0, ∞) to satisfy Pn(t) = 0 such that Pn(t) < 0 for t ∈ (0, tm+1) and Pn(t) > 0 for t ∈ (tm+1, ∞).

Consequently, for given t0 > 0, if Pn(t0) > 0 then Pn(t) > 0 for t ∈ (t0, ∞) and if Pn(t0) < 0 then Pn(t) < 0 for t ∈ (0, t0).

#### Lemma 3.3

Let W01(x), $\begin{array}{}{W}_{01}^{\ast }\end{array}$ (x), W1(x), Wc1(x) and Wl1(x) be defined by (19), (20), (6), (21) and (22), respectively. Then we have

$W1x

for all x ≥ 1.

#### Proof

1. The first inequality W1(x) < Wc1(x) is equivalent to

$h1t=lnsinh⁡tt+1810t6−1+1135t4lnsinh⁡tt<0$

for t = 1/x ∈ (0, 1]. We have

$ddylny+1810t6−1+1135t4ln⁡y=−1135t4810y+135t2+t6y810y+t6<0$

for y > 1, which together with

$y=sinh⁡tt>1+16t2$

yields

$h1t

$1352t3h11′t=−2ln16t2+1−t2t6−135t2−1620t2+6t6+135t2+810:=h12t,h12′t=−4t3t12+9t10+540t8+8505t6+47385t4+328050t2+1312200t2+62t6+135t2+8102<0$

for t > 0. Therefore, we obtain h12(t) < h12(0) = 0, and so h11(t) < h11(0) = 0, which implies h1(t) < 0 for t > 0.

2. The second inequality Wc1(x) < $\begin{array}{}{W}_{01}^{\ast }\end{array}$ (x) is equivalent to

$x21+1135x4lnxsinh⁡1x

or equivalently,

$h2t=1270t3lnsinh⁡tt−ln1+11620t5<0$

for t = 1/x ∈ (0, 1]. Taking n = 2 in the inequalities (13) gives

$lnsinh⁡tt<16t2−1180t4+12835t6,$

which is applied to the expression of h2(t):

$h2t<1270t316t2−1180t4+12835t6−ln1+11620t5:=h21t.$

Differentiation yields

$h21′t=t63402004t7−49t5+1050t3+6480t2−79380t5+1620<0$

for t ∈ (0, 1], which proves h2(t) < 0 for t ∈ (0, 1].

3. The third inequality $\begin{array}{}{W}_{01}^{\ast }\end{array}$ (x) < W01(x) is equivalent to

$1+11620x5

which follows by a simple inequality 1 + y < ey for y ≠ 0.

4. The fourth inequality W01(x) < Wl1(x) is equivalent to

$x2lnxsinh1x+1810x7>x2lnxsinh⁡1x+11620x5,$

or equivalently,

$h3t=lnsinht+1810t7−lnsinh⁡t−1810t6>0$

for t = 1/x > 0. Denote h30(t) = ln sinh t. Then by Taylor formula we have

$h3t=h30t+1810t7−h30t−1810t6 =t7810h30′t+12!t148102h30′′t+13!t218103h30′′′ξ−1810t6,$

where t < ξ < t + t7/810. Since $\begin{array}{}{h}_{30}^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}\end{array}$ (t) = 2(cosh t)/sinh3t > 0, we get

$h3t>1810t7cosh⁡tsinh⁡t−t142×81021sinh2⁡t−1810t6:=t6×h31t2×8102sinh2⁡t,$

where

$h31t=810tsinh⁡2t−810cosh⁡2t+810−t8.$

Due to

$h31t=540t4+144t6+1017t8+810∑n=5∞n−122n2n!t2n>0,$

we conclude that h3(t) > 0 for t > 0, which completes the proof. □

## 3.3 The comparison theorem for W1(x), Wc1(x), $\begin{array}{}{W}_{01}^{\ast }\end{array}$ (x), W01(x) and Wl1(x)

#### Theorem 3.4

1. The function

$f1x=lnΓx+1−ln⁡2π−x+12ln⁡x+x−x2lnxsinh⁡1x+1810x6$

is strictly increasing and concave on [1, ∞).

2. For x ≥ 1, we have

$β0xsinh⁡1x+1810x6x/2<Γx+12πxx/ex(25)

with the best constant

$β0=e2πsinh⁡1+π/405≈0.99981.$

#### Proof

1. Differentiation yields

$f1′x=ψx+1−ln⁡x−12x−12lnxsinh⁡1x+1810x6 +3135x6cosh⁡1x−135x7sinh⁡1x+1810x7sinh⁡1x+1,$

$f1′′x=ψ′x+1−1x+12x2−3109350x14sinh2⁡1x+5940x7sinh⁡1x+135x5sinh⁡1x−1890x6cosh⁡1x−109350x12−1x810x7sinh⁡1x+12.$

Replacing x by x + 1/2 in inequality (23) yields

$ψ′x+1<1303780x4+7560x3+12705x2+8925x+30192x+163x4+126x3+217x2+154x+60,$

for x > − 1/2, and and applying the above inequality combined with the change of variable x = 1/t ∈ (0, 1] yield

$f1′′x<130t3019t4+8925t3+12705t2+7560t+3780t+260t4+154t3+217t2+126t+63−t+12t2−3t109350sinh2⁡t−1890t8cosh⁡t+5940t7sinh⁡t+135t9sinh⁡t−109350t2−t14810sinh⁡t+t72:=8102t×f11tt+2126t+217t2+154t3+60t4+63810sinh⁡t+t72,$

where

$f11t=p6tsinh2⁡t+p13tcosh⁡t−p14tsinh⁡t+p20t, p6t=30t6+47t5−71815t4−210t3−259t2−3152t−63, p13t=7810t8t+260t4+154t3+217t2+126t+63,p14t=t7127t7+77810t6+28571620t5+459736075t4+1547108t3+12341810t2+779t+15445, p20t=121870t20+257656100t19+136679841500t18+21787480t17+72700t16+74860t15+712150t14+30t7+137t6+5252t5+280t4+3152t3+63t2.$

To prove f11(t) < 0 for t ∈ (0, 1], we use formula sinh2t = cosh2t − 1 to write f11(t) as

$f31t=p6tcosh⁡t+p13tcosh⁡t−p14tsinh⁡t+p20t−p6t.$

Since the coefficients of polynomial − p6(t) satisfy those conditions of Lemma 3.2, and − p6(1) = 19 811/30 > 0, we see that − p6(t) > 0 for t ∈ (0, 1]. It then follows from that

$p6tcosh⁡t+p13t

Application of Lemma 3.2 again with $\begin{array}{}-{p}_{13}^{\ast }\end{array}$ (1) = 173 959/270 > 0 yields $\begin{array}{}-{p}_{13}^{\ast }\end{array}$ (t) > 0 for t ∈ (0, 1], and so p6(t) cosh t + p13(t) < 0 for t ∈ (0, 1]. Since p14(t) > 0 for t > 0, using the inequalities

$cosh⁡t>∑n=04t2n2n!=140320t8+1720t6+124t4+12t2+1,sinh⁡t>∑n=14t2n−12n−1!=15040t7+1120t5+16t3+t,$

we have

$f11t=p6tcosh⁡t+p13tcosh⁡t−p14tsinh⁡t+p20t−p6t

From Lemma 3.2 and − p12(1) = 67 766 507 802 179/3950 456 832 000 > 0 it follows that − p12(t) > 0 for t ∈ (0, 1], and so f11(t) < 0 for t ∈ (0, 1], which implies $\begin{array}{}{f}_{1}^{\mathrm{\prime }\mathrm{\prime }}\end{array}$ (x) < 0 for x ≥ 1.

2. Using the increasing property of f1 and noting that

$f11=ln⁡e2πsinh⁡1+π/405 and limx→∞f1x=0,$

we have

$ln⁡e2πsinh⁡1+π/405

which imply the first and second inequalities of (25).

The other ones of (25) follow from Lemma 3.3, which completes the proof. □

## 4 Conclusions

In this paper, by a little known power series expansion of ln (t−1 sinh t), that is, (12), we establish an asymptotic expansion (10) for the gamma function related to Windschitl’s formula, in which its coefficients have a closed-form expression (14). Moreover, we give an estimate of the remainder in the asymptotic expansion (10) by means of inequalities (3) and (13). Due to (12), we also give other two asymptotic expansions (11) and (18), but their coefficients formulas are of recursive form. Despite all that, the latter improves Chen’s result [34, Theorem 2]

Furthermore, we compare the accuracy of all five approximation formulas for the gamma function generated by truncating five asymptotic series (10), (11), (9), (18) and (7) by numeric computations and some inequalities. These show that the approximation formula (6) is the best. Some general properties of truncated series (truncated polynomials) can refer to [46].

## Acknowledgement

The authors would like to express their sincere thanks to the anonymous referees for their great efforts to improve this paper.

This work was supported by the Fundamental Research Funds for the Central Universities (No. 2015ZD29) and the Higher School Science Research Funds of Hebei Province of China (No. Z2015137).

## References

• [1]

Ramanujan S., The Lost Notebook and Other Unpublished Papers, 1988, Berlin: Springer. Google Scholar

• [2]

Burnside W., A rapidly convergent series for log N!, Messenger Math., 1917, 46, 157–159. Google Scholar

• [3]

Gosper R. W., Decision procedure for indefinite hypergeometric summation, Proc. Nat. Acad. Sci. U.S.A., 1978, 75, 40–42.

• [4]

Alzer H., On some inequalities for the gamma and psi functions, Math. Comp., 1997, 66, 373–389.

• [5]

Alzer H., Sharp upper and lower bounds for the gamma function, Proc. Roy. Soc. Edinburgh Sect. A, 2009, 139, 709–718.

• [6]

Smith W. D., The gamma function revisited, http://schule.bayernport.com/gamma/gamma05. pdf, 2006.

• [7]
• [8]

Batir N., Sharp inequalities for factorial n, Proyecciones, 2008, 27, 97–102. Google Scholar

• [9]

Batir N., Inequalities for the gamma function, Arch. Math., 2008, 91, 554–563.

• [10]

Mortici C., An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math., 2009, 93, 37–45.

• [11]

Mortici C., New sharp inequalities for approximating the factorial function and the digamma functions, Miskolc Math. Notes, 2010, 11, 79–86.

• [12]

Mortici C., A new Stirling series as continued fraction, Numer. Algor., 2011, 56, 17–26.

• [13]

Mortici C., Improved asymptotic formulas for the gamma function, Comput. Math. Appl., 2011, 61, 3364–3369.

• [14]

Mortici C., Further improvements of some double inequalities for bounding the gamma function, Math. Comput. Model., 2013, 57, 1360–1363.

• [15]

Mortici C., A continued fraction approximation of the gamma function, J. Math. Anal. Appl., 2013, 402, 405–410.

• [16]

Nemes G., New asymptotic expansion for the Gamma function, Arch. Math. (Basel), 2010, 95, 161–169.

• [17]

Nemes G., More accurate approximations for the gamma function, Thai J. Math., 2011, 9, 21–28. Google Scholar

• [18]

Guo B.-N., Zhang Y.-J., Qi F., Refinements and sharpenings of some double inequalities for bounding the gamma function, J. Inequal. Pure Appl. Math., 2008, 9, Article 17. Google Scholar

• [19]

Qi F., Integral representations and complete monotonicity related to the remainder of Burnside’s formula for the gamma function, J. Comput. Appl. Math., 2014, 268, 155–167.

• [20]

Chen Ch.-P., A more accurate approximation for the gamma function, J. Number Theory, 2016, 164, 417–428.

• [21]

Yang Zh.-H., Tian J.-F., Monotonicity and inequalities for the gamma function, J. Inequal. Appl., 2017, 2017, 317.

• [22]

Yang Zh.-H., Tian J.-F., An accurate approximation formula for gamma function, J. Inequal. Appl., 2018, 2018, 56.

• [23]

Yang Zh.-H., Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function, J. Math. Anal. Appl., 2016, 441, 549–564.

• [24]

Lu D., A new sharp approximation for the Gamma function related to Burnside’s formula, Ramanujan J., 2014, 35, 121–129.

• [25]

Lu D., Song L., Ma C., Some new asymptotic approximations of the gamma function based on Nemes’ formula, Ramanujan’s formula and Burnside’s formula, Appl. Math. Comput., 2015, 253, 1–7.

• [26]

Abramowttz M., Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 1972, New York: Dover. Google Scholar

• [27]

Koumandos S., Remarks on some completely monotonic functions, J. Math. Anal. Appl., 2006, 324, 1458–1461.

• [28]

Chen C.-P., Lin L., Remarks on asymptotic expansions for the gamma function, Appl. Math. Lett., 2012, 25, 2322–2326.

• [29]

Chen C.-P., Elezović N., Vukšić L., Asymptotic formulae associated with the Wallis power function and digamma function, J. Classical Anal., 2013, 2, 151–166. Google Scholar

• [30]

Chen C.-P., Unified treatment of several asymptotic formulas for the gamma function, Numer. Algor., 2013, 64, 311–319.

• [31]

Lu D., A generated approximation related to Burnside’s formula, J. Number Theory, 2014, 136, 414–422.

• [32]

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.

• [33]

Hirschhorn M. D., Villarino M. B., A refinement of Ramanujan’s factorial approximation, Ramanujan J., 2014, 34, 73–81.

• [34]

Chen C.-P., Asymptotic expansions of the gamma function related to Windschitl’s formula, Appl. Math. Comput., 2014, 245, 174–180.

• [35]

Chen C.-P., Liu J.-Y., Inequalities and asymptotic expansions for the gamma function, J. Number Theory, 2015, 149, 313–326.

• [36]

Chen C.-P., Paris R. B., Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function, Appl. Math. Comput., 2015, 250, 514–529.

• [37]

Lin L., Chen C.-P., Asymptotic formulas for the gamma function by Gosper, J. Math. Inequal., 2015, 9, 541–551.

• [38]

Mortici C., A new fast asymptotic series for the gamma function, Ramanujan J., 2015, 38, 549–559.

• [39]

Yang Zh., Tian J.-F., A comparison theorem for two divided differences and applications to special functions, J. Math. Anal. Appl., 2018, 464, 580–595.

• [40]

Tian J., Wang W., Cheung W.-S., Periodic boundary value problems for first-order impulsive difference equations with time delay, Adv. Difference Equ., 2018, 2018, 79.

• [41]

Tian J.-F., Triple Diamond-Alpha integral and Hölder-type inequalities, J. Inequal. Appl., 2018, 2018, 111. Google Scholar

• [42]

Tian J.-F., Ha M.-H., Wang Ch., Improvements of generalized Hölder’s inequalities and their applications, J. Math. Inequal., 2018, 12, 459–471. Google Scholar

• [43]

Yang Zh.-H., Chu Y.-M., Tao X.-J, A double inequality for the trigamma function and its applications, Abstr. Appl. Anal., 2014, 2014, Art. ID 702718, 9 pages.

• [44]

Yang Zh.-H., Tian J., Monotonicity and sharp inequalities related to gamma function, J. Math. Inequal., 2018, 12, 1–22.

• [45]

Yang Zh.-H., Tian J., Convexity and monotonicity for the elliptic integrals of the first kind and applications, arXiv:1705.05703 [math.CA], https://arxiv.org/abs/1705.05703

• [46]

Dattoli G., Cesarano C., Sacchetti D., A note on truncated polynomials, Appl. Math. Comput., 2003, 134, 595–605. Google Scholar

Accepted: 2018-07-17

Published Online: 2018-08-24

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1048–1060, ISSN (Online) 2391-5455,

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