Journal of Number Theory 140 (2014) 314–323

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Journal of Number Theory

www.elsevier.com/locate/jnt

A new asymptotic expansion and some inequalities for thegamma function

Dawei Lu ∗, Xiaoguang WangSchool of Mathematical Sciences, Dalian University of Technology, Dalian 116023, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 December 2013Received in revised form 21 January2014Accepted 31 January 2014Available online 5 March 2014Communicated by David Goss

MSC:33B1541A1042A16

Keywords:Gamma functionContinued fractionInequalitiesBurnside formula

In this paper, based on the Burnside formula, an asymptoticexpansion of the factorial function and some inequalitiesfor the gamma function are established. Finally, for demon-strating the superiority of our new series over the Burnsideformula, the classical Stirling series and the Mortici sequences,some numerical computations are given.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction

It is well known that we often need to deal with the big factorials in many situationsin pure mathematics and other branches of science. The Stirling formula

n! ≈√

2πn(n

e

)n

(1.1)

* Corresponding author.

E-mail addresses: ludawei_dlut@163.com (D. Lu), wangxg@dlut.edu.cn (X. Wang).

http://dx.doi.org/10.1016/j.jnt.2014.01.0250022-314X/© 2014 Elsevier Inc. All rights reserved.

D. Lu, X. Wang / Journal of Number Theory 140 (2014) 314–323 315

is one of the most known formulas for approximation of the factorial function. Up to now,many researchers made great efforts in the area of establishing more precise inequalitiesand more accurate approximations for the factorial function and its extension gammafunction, and had a lot of inspiring results. For example, the Stirling series [1],

n! ≈√

2πn(n

e

)n(1 + 1

12n + 1288n2 − 139

51 840n3 − 5712 488 320n4 + · · ·

), (1.2)

is, in fact, an extension of (1.1). The Burnside formula [3],

n! ≈√

2π(n + 1

2e

)n+ 12

, (1.3)

is more precise than (1.1). There are also two approximations which are better than(1.3), the Gosper formula [4],

n! ≈√

2π(n + 1

6

)(n

e

)n

, (1.4)

and the Ramanujan formula [14],

n! ≈√

2π(n

e

)n(n3 + 1

2n2 + 1

8n + 1240

)1/6

. (1.5)

The more exact formula without simple shape is the Windschitl formula [15],

n! ≈√

2πn(n

e

)n(n sinh 1

n

)n/2

. (1.6)

The Nemes formula [13],

n! ≈√

2πn(n

e

)n(1 + 1

12n2 − 110

)n

, (1.7)

has the same number of exact digits as (1.6) but is much simpler.Recently, some authors also paid attention to giving increasingly better approxima-

tions for the gamma function using continued fractions. For example, using an ownmethod, Mortici rediscovered in [11] the Stieltjes continued fraction

Γ (x + 1) ≈√

2πx(x

e

)x

exp(

a0

x + a1x+ a2

x+···

), (1.8)

where

a0 = 1, a1 = 1

, a2 = 53 etc.

12 30 210

316 D. Lu, X. Wang / Journal of Number Theory 140 (2014) 314–323

In addition, Mortici [12] also provided a new continued fraction approximation startingfrom the Nemes formula (1.7) as follows,

Γ (x + 1) ≈√

2πxe−x

⎛⎜⎜⎜⎜⎜⎜⎜⎝x + 1

12x− 110x+ a

x+ bx+ c

x+ d

x+. . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

x

, (1.9)

where

a = −2369252 , b = 2 117 009

1 193 976 , c = 393 032 191 5111 324 011 300 744 ,

d = 33 265 896 164 277 124 002 45114 278 024 104 089 641 878 840 · · · .

It is their works that motivate our study. In this paper, based on the early works ofMortici, using the same idea from (1.1) to (1.9), we provide a new asymptotic expansionfor the factorial function starting from the Burnside formula (1.3) as follows:

Theorem 1.1. For the factorial function, we have

n! ≈√

2π(n + 1

2e

)n+ 12

⎛⎜⎜⎜⎜⎝1 + a1

n2 + a2nn+ a3n

n+ a4n

n+. . .

⎞⎟⎟⎟⎟⎠

n− 12

, (1.10)

where

a1 = − 124 , a2 = − 41

240 , a3 = −1441 , a4 = 1 695 131

2 892 960 · · · .

It is easy to see that the Burnside formula (1.3) is only a particular case of (1.10) forall ai = 0, i � 1. Comparing with the continued fraction approximations (1.9), the newasymptotic expansion (1.10) has similar shape.

Next, using Theorem 1.1, we provide some inequalities for the gamma function.

Theorem 1.2. There exists an m, such that for every x � 1, it holds:

(1 −

124

x2 −41240x

x− 1441

)x− 12

<Γ (x + 1)

√2π(x+ 1

2e )x+ 1

2

<

(1 −

124

x2 − 41240

)x− 12

. (1.11)

D. Lu, X. Wang / Journal of Number Theory 140 (2014) 314–323 317

To obtain Theorem 1.1, we need the following lemma which was used in [5–12] andis very useful for constructing asymptotic expansions and for accelerating some conver-gences.

Lemma 1.1. If (xn)n�1 is convergent to zero and there exists the limit

limn→∞

ns(xn − xn+1) = l ∈ (−∞,+∞), (1.12)

with s > 1, then

limn→∞

ns−1xn = l

s− 1 . (1.13)

Lemma 1.1 was first proved by Mortici in [10]. From Lemma 1.1, we can see thatthe speed of convergence of the sequence (xn)n�1 increases together with the value s

satisfying (1.12).The rest of this paper is arranged as follows. In Section 2, we provide the proof

of Theorem 1.1. In Section 3, the proof of Theorem 1.2 is given. In Section 4, wegive some numerical computations which demonstrate the superiority of our newseries over the Burnside formula, the classical Stirling series and the Mortici se-quences.

2. Proof of Theorem 1.1

Based on the argument of Theorem 2.1 in [11] or Theorem 5 in [12], we need to findthe value a1 ∈ R which produces the most accurate approximation of the form

n! ≈√

2π(n + 1

2e

)n+ 12(

1 + a1

n2

)n− 12

. (2.1)

To measure the accuracy of this approximation, a method is to define a sequence (tn)n�1

by the relations

n! =√

2π(n + 1

2e

)n+ 12(

1 + a1

n2

)n− 12

exp(tn) (2.2)

and to say that an approximation (2.1) is better if tn converges to zero faster.From (2.2), we have

tn = lnn! − 12 ln 2π −

(n + 1

2

)ln(n + 1

2

)+ n + 1

2 −(n− 1

2

)ln(

1 + a1

n2

). (2.3)

318 D. Lu, X. Wang / Journal of Number Theory 140 (2014) 314–323

Thus,

tn − tn+1 = −1 +(n + 1

2

)ln(

1 + 1n + 1

2

)+ ln

(1 +

12

n + 1

)

−(n− 1

2

)ln(

1 + a1

n2

)+

(n + 1

2

)ln(

1 + a1

(n + 1)2

). (2.4)

Developing tn − tn+1 in a power series in 1/n, we have

tn − tn+1 = −(

124 + a1

)1n2 +

(112 + 2a1

)1n3 +

(32a

21 −

52a1 −

41320

)1n4 + O

(1n5

).

(2.5)

From Lemma 1.1, we know that the speed of convergence of the sequence (tn)n�1 is evenhigher as the value s satisfying (1.12). Thus, using Lemma 1.1, we have,

(i) If a1 �= − 124 , then the rate of convergence of the sequence (tn)n�1 is n−1 since

limn→∞

ntn = −(

124 + a1

)�= 0.

(ii) If a1 = − 124 , then from (2.5), we have

tn − tn+1 = − 411920

1n4 + O

(1n5

)

and the rate of convergence of the sequence (tn)n�1 is n−3 since

limn→∞

n3tn = − 415760 .

We know that the fastest possible sequence (tn)n�1 is obtained only for a1 = −1/24.Next, we define the sequence (wn)n�1 by the relation

n! =√

2π(n + 1

2e

)n+ 12(

1 +− 1

24n2 + a2

)n− 12

exp(wn), (2.6)

where a2 is any real number. Using the same method from (2.1) to (2.5), we have thatthe fastest possible sequence (wn)n�1 is obtained only for a2 = −41/240.

Then, we define the sequence (vn)n�1 by the relation

n! =√

2π(n + 1

2e

)n+ 12(

1 +− 1

24

n2 + − 41240n

)n− 12

exp(vn). (2.7)

n+a3

D. Lu, X. Wang / Journal of Number Theory 140 (2014) 314–323 319

Using the same method from (2.1) to (2.5), we have that the fastest possible sequence(vn)n�1 is obtained only for a3 = −14/41.

Similar proceeding, we get a4 = 1 695 1312 892 960 , . . . , and the new asymptotic expansion (1.10)

is obtained.

3. Proof of Theorem 1.2

To give the proof of Theorem 1.2, we first need the following basic result of Alzer [2]for all x > 0 and n � 0,

lnΓ (x + 1) =(x + 1

2

)ln x− x + 1

2 ln 2π +n∑

j=1

B2j

2j(2j − 1)x2j−1 + (−1)nRn(x), (3.1)

where Rn(x) is completely monotonic on (0,∞), Bj is the jth Bernoulli number definedby the power series expansion

x

ex − 1 =∞∑l=0

Blxl

l! = 1 − x

2 +∞∑l=1

B2lx2l

(2l)! . (3.2)

(B2l+1 = 0, for all l � 1, and the first few Bernoulli numbers are B1 = −1/2, B2 = 1/6,B4 = −1/30, B6 = 1/42.) From (3.2), Mortici obtained the following inequalities in [11]and [12], for x > 0,

exp(

112x − 1

360x3 + 11260x5 − 1

1680x7

)

<Γ (x + 1)√2πx(xe )x

< exp(

112x − 1

360x3 + 11260x5

). (3.3)

For the upper bound in inequality (1.11), combining (3.3), we need to get

exp(

12 + 1

12x − 1360x3 + 1

1260x5

)

<

(1 + 1

2x

)x+ 12(

1 −124

x2 − 41240

)x− 12

. (3.4)

Inequality (3.4) is equivalent to f(x) < 0, where

f(x) = 12 + 1

12x − 1360x3 + 1

1260x5 −(x + 1

2

)ln(

1 + 12x

)

−(x− 1

2

)ln(

1 −124

x2 − 41240

). (3.5)

From (3.5), we have

f ′′(x) = − A(x− 1)7 2 2 2 2 , (3.6)

210x (2x + 1)(240x − 41) (80x − 17)

320 D. Lu, X. Wang / Journal of Number Theory 140 (2014) 314–323

where

A(x) = 16 194 377 409x + 94 669 008 621x2 + 324 310 766 496x3 + 714 487 613 580x4

+ 1 051 623 619 955x5 + 1 044 151 969 600x6 + 690 060 667 200x7

+ 290 726 272 000x8 + 70 622 240 000x9 + 7 526 400 000x10 + 1 247 978 214.

(3.7)

f ′′ < 0, so f is strictly concave. As limx→∞ f(x) = 0, we get f < 0 on [1,∞).For the lower bound in (1.11), combining (3.3), we need to get

(1 + 1

2x

)x+ 12(

1 −124

x2 −41240x

x− 1441

)x− 12

< exp(

12 + 1

12x − 1360x3 + 1

1260x5 − 11680x7

). (3.8)

Inequality (3.8) is equivalent to g(x) > 0, where

g(x) = 12 + 1

12x − 1360x3 + 1

1260x5 − 11680x7 −

(x + 1

2

)ln(

1 + 12x

)

−(x− 1

2

)ln(

1 −124

x2 −41240x

x− 1441

). (3.9)

From (3.9), we have

g′′(x) = B(x− 1)210x9(2x + 1)(9840x3 − 3360x2 − 2091x + 140)2(9840x2 − 3360x− 1681)2 ,

(3.10)

where

B(x) = 287 403 924 498 943 399x + 2 168 352 592 121 458 362x2

+ 10 053 814 706 617 707 616x3 + 31 824 430 481 543 004 300x4

+ 72 131 334 522 408 857 224x5 + 119 574 747 664 100 601 135x6

+ 145 928 956 123 876 311 286x7 + 130 586 201 334 200 124 840x8

+ 84 441 489 745 982 305 605x9 + 38 319 227 909 536 987 200x10

+ 11 556 824 842 016 107 200x11 + 2 077 411 857 880 320 000x12

+ 168 235 378 057 440 000x13 + 17 687 016 380 767 293. (3.11)

g′′ > 0, so g is strictly convex. As limx→∞ g(x) = 0, we get g > 0 on [1,∞).

D. Lu, X. Wang / Journal of Number Theory 140 (2014) 314–323 321

Fig. 1. L(x) = Solid and U(x) = Dot.

Comments. In fact, let us take

L(x) = Γ (x + 1) −√

2π(x + 1

2e

)x+ 12(

1 −124

x2 −41240x

x− 1441

)x− 12

,

U(x) =√

2π(x + 1

2e

)x+ 12(

1 −124

x2 − 41240

)x− 12

− Γ (x + 1).

By some simulations, we obtain Fig. 1. From Fig. 1, it is easy to see that L(x) > 0 andU(x) > 0 uniformly on [1,∞). Thus, we consider that it is suitable to take m = 1 inTheorem 1.2.

4. Numerical computations

In this section, we give a comparison table to demonstrate the superiority of our newseries

n! ≈√

2π(n + 1

2e

)n+ 12(

1 −124

n2 −41240n

n− 1441

)n− 12

= γn (4.1)

over the Burnside formula

n! ≈√

2π(n + 1

2e

)n+ 12

= βn (4.2)

the classical Stirling series truncated at the n−3 term,

322 D. Lu, X. Wang / Journal of Number Theory 140 (2014) 314–323

Table 1Simulations for βn, νn, μn, θn and γn.

n (βn − n!)/n! (νn − n!)/n! (n! − μn)/n! (n! − θn)/n! (n! − γn)/n!

50 8.2540 × 10−4 3.4145 × 10−11 2.1213 × 10−10 5.8555 × 10−10 4.5707 × 10−12

100 4.1467 × 10−4 2.2144 × 10−12 6.6870 × 10−12 7.9957 × 10−11 1.4263 × 10−13

500 8.3253 × 10−5 3.6458 × 10−15 2.1549 × 10−15 6.8343 × 10−13 4.5587 × 10−17

2500 1.6663 × 10−5 5.8662 × 10−18 6.7401 × 10−17 8.6117 × 10−14 1.4584 × 10−20

n! ≈√

2πn(n

e

)n(1 + 1

12n + 1288n2 − 139

51 840n3

)= νn, (4.3)

the Mortici sequences in [8,9]

n! ≈√

2πe

(n + 1e

)n+ 12

exp(

112n − 1

12n2 + 29360n3 − 3

40n4

)= μn, (4.4)

n! ≈√

2πe

(n + 1

2e

)n(n3 + 5x2

4 + 17x32 + 172

1920

)1/6

= θn, (4.5)

respectively. Combining Theorem 1.2, we have Table 1.

Acknowledgments

We thank the referees for careful reading of our manuscript and for helpful andvaluable comments and suggestions. The comments by the referees helped the authorsimprove the exposition of this paper significantly. The research of the first author is Sup-ported by the National Natural Science Foundation of China (grant numbers 11101061and 11371077), Research Foundation for Doctor of Liaoning Province (grant number20121016) and the Fundamental Research Funds for the Central Universities (grantnumbers DUT12LK16 and DUT13JS06). The second author is supported by the Na-tional Natural Science Foundation of China (grant numbers 11101063 and 61173103),the Fundamental Research Funds for the Central Universities in China (DUT12LK29).

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