Some new asymptotic approximations of the ... - Semantic Scholar
Applied Mathematics and Computation 253 (2015) 1–7
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some new asymptotic approximations of the gamma function based on Nemes’ formula, Ramanujan’s formula and Burnside’s formula Dawei Lu, Lixin Song ⇑, Congxu Ma School 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 In this paper, we construct some new approximations of the gamma function based on Nemes’ formula, Ramanujan’s formula and Burnside’s formula. Using these approximations, some inequalities are established. Finally, for demonstrating the superiority of our new approximation over Mortici’s formula and other classical ones, some numerical computations are also given. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Gamma function Nemes’ formula Ramanujan’s formula Burnside’s formula Asymptotic approximations Rate of convergence
1. Introduction We often need to deal with big factorials in many areas of mathematics and science in general. Undoubtedly, one of the most known formulas for approximation of the factorial function is Stirling’s formula [1],
n!
pffiffiffiffiffiffiffiffiffinn : 2pn e
ð1:1Þ
The next step in this direction is to define more accurate approximations for the factorial function. A more accurate approximation than formula (1.1) is Burnside’s formula [3],
pffiffiffiffiffiffiffin þ 1nþ2 2 : 2p e 1
n!
ð1:2Þ
There are two approximations which are better than (1.2), Gosper’s formula [4],
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 nn n! 2p n þ ; 6 e
ð1:3Þ
and Ramanujan’s formula [5],
n!
pffiffiffiffiffiffiffinn 2p e
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 1 1 1 n3 þ n2 þ n þ : 2 8 240
Windschitl’s formula [6],
⇑ Corresponding author. E-mail addresses: [email protected] (D. Lu), [email protected] (L. Song), [email protected] (C. Ma). http://dx.doi.org/10.1016/j.amc.2014.12.077 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð1:4Þ
2
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
n!
n=2 pffiffiffiffiffiffiffiffiffinn 1 n sinh ; 2pn e n
ð1:5Þ
is more exact. Nemes’ formula [7],
!n pffiffiffiffiffiffiffiffiffinn 1 n! 2pn 1þ ; 1 e 12n2 10
ð1:6Þ
has the same number of exact digits as (1.5) but is much simpler. Recently, many researchers presented some more accurate approximations for the factorial function and its extension gamma function. Lu [14] presented a polynomial approximation for the factorial function as follows,
n!
1=k pffiffiffiffiffiffiffiffiffinn c1 c2 c3 1 þ þ 2 þ 3 þ ; 2pn e n n n
c1 ¼
k ; 12
ð1:7Þ
where 2
c2 ¼
k ; 288
2
c3 ¼
kð5k 144Þ ; 51840
Based on the works of Windschitl and Mortici, Lu [15] provided a formula which is an extension of Windschitl’s formula as follows,
n!
n=2 pffiffiffiffiffiffiffiffiffinn 1 a7 a9 a11 n sinh þ 7 þ 9 þ 11 þ ; 2pn e n n n n
a7 ¼
1 ; 810
ð1:8Þ
where
a9 ¼
67 ; 42525
a11 ¼
19 ; 8505
In addition, some authors paid attention to giving increasingly better approximations for the gamma function by using the continued fractions. For example, Mortici [13] provided a new continued fraction approximation starting from Nemes’ formula (1.4) as follows,
1x
0
Cðx þ 1Þ
B pffiffiffiffiffiffiffiffiffi B 1 B 2pxex Bx þ B 12x 10xþ 1 @ xþ
a b
c xþ xþ d . xþ . .
C C C C; C A
ð1:9Þ
where
a¼
2369 ; 252
b¼
2117009 ; 1193976
c¼
393032191511 ; 1324011300744
d¼
33265896164277124002451 ; 14278024104089641878840
Using the continued fractions, Lu [16] provided an asymptotic expansion for the factorial function starting from Burnside’s formula (1.2) as follows,
0 1 xþ 2B pffiffiffiffiffiffiffi x þ 1 a1 B 2 Cðx þ 1Þ 2p B1 þ 2 @ e x þ a2ax3 x xþ
a x xþ 4 . xþ . .
1x12 C C C A
;
ð1:10Þ
where
a1 ¼
1 ; 24
a2 ¼
41 ; 240
a3 ¼
14 ; 41
a4 ¼
1695131 ; 2892960
Very recently, using a different method, Chen [17] gave a new approximation related to Windschitl’s formula,
Cðx þ 1Þ
xþ 1 þ 191 þ 25127 þ pffiffiffiffiffiffiffiffiffixx 1 2 270x3 56700x5 5103000x7 x sinh : 2px e x
ð1:11Þ
It is their works that motivate our study. In this paper, based on Nemes’ formula (1.6), Ramanujan’s formula (1.4), the extension of Burnside’s formula (1.10) and the works of Chen [17], using the same method in Mortici [13], we introduce some new approximations as follows,
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
3
Theorem 1.1. For the factorial function, based on (1.6), we have
n!
!nþ 2369 3 6157 5 þ 14162903 7 þ 907200n 1437004800n 302400n pffiffiffiffiffiffiffiffiffinn 1 2pn 1þ : 1 2 e 12n 10
ð1:12Þ
Based on (1.4), we have
n!
16 11 3 þ 13 4 7 5 þ pffiffiffiffiffiffiffiffiffinn 115200n 5760n 6720n 1 1 1 1þ þ 2þ : 2pn e 2n 8n 240n3
ð1:13Þ
Based on (1.10), we have
!n12þ 7 2 52531 3 þ 6763 4 þ 1 1 120n 1209600n 2419200n pffiffiffiffiffiffiffin þ 1nþ2 24 2 n! 2p 1 þ 2 41 : e n 240
ð1:14Þ
Next, using Theorem 1.1, we provide some inequalities for the gamma function. Theorem 1.2. (i) For every x > 2, it holds:
!xþ 2369 3 6157 5 907200x 302400x 1 Cðx þ 1Þ 1þ < pffiffiffiffiffiffiffiffiffixx < 1 2 12x 10 2px e
!xþ 2369 3 302400x 1 1þ : 1 2 12x 10
ð1:15Þ
(ii) For every x > 2, it holds:
16 11 3 16 11 3 þ 13 4 5760x 5760x 6720x 1 1 1 Cðx þ 1Þ 1 1 1 p ffiffiffiffiffiffiffiffi ffi 1þ þ 2þ < < 1 þ þ þ : 2 3 x x 2x 8x 240x3 2x 8x 240x 2px e
ð1:16Þ
(iii) For every x > 2, it holds:
!x12þ 7 2 1 120x 24 Cðx þ 1Þ 1 þ 2 41 < < pffiffiffiffiffiffiffixþ1xþ12 x 240 2p e 2
!x12þ 7 2 52531 3 1 120x 1209600x 24 1 þ 2 41 : x 240
ð1:17Þ
To obtain Theorem 1.1, we need the following lemma which was used in [8–10,13] and is very useful for constructing asymptotic expansions. Lemma 1.1. If ðxn ÞnP1 is convergent to zero and there exists the limit
lim ns ðxn xnþ1 Þ ¼ l 2 ½1; þ1;
n!1
ð1:18Þ
with s > 1, then
lim ns1 xn ¼
n!1
l : s1
ð1:19Þ
Lemma 1.1 was first proved by Mortici in [11]. From Lemma 1.1, we can see that the speed of convergence of the sequence ðxn ÞnP1 increases together with the value s satisfying (1.18). The rest of paper is arranged as follows. In Section 2, the proof of Theorem 1.1 is given. In Section 3, we provide the proof of Theorem 1.2. In Section 4, we give some numerical computations to demonstrate the superiority of our new series respectively. 2. Proof of Theorem 1.1 Based on the argument of Theorem 2.1 in [12] or Theorem 5 in [13], we need to find the value a1 2 R which produces the most accurate approximation of the form
!nþa13 n pffiffiffiffiffiffiffiffiffinn 1 n! 2pn 1þ : 1 e 12n2 10
ð2:1Þ
To measure the accuracy of this approximation, a method is to define a sequence ðt n ÞnP1 by the relations
!nþa13 n pffiffiffiffiffiffiffiffiffinn 1 n! 2pn 1þ expðt n Þ; 1 2 e 12n 10
ð2:2Þ
4
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
and to say that an approximation (2.1) is better if t n converges to zero faster. From (2.2), we have
! 1 1 a1 1 : ln n þ n n þ 3 ln 1 þ tn ¼ ln n! ln 2p n þ 1 2 2 n 12n2 10
ð2:3Þ
Thus,
tn t nþ1
! 1 1 a1 1 ln 1 þ n þ 3 ln 1 þ ¼ 1 þ n þ 1 2 n n 12n2 10 ! ! a1 1 ln 1 þ : þ nþ1þ 1 12ðn þ 1Þ2 10 ðn þ 1Þ3
ð2:4Þ
Developing the power series in 1=n, we have
tn t nþ1 ¼
302400a1 þ 2369 302400a1 þ 2369 1668800a1 þ 10793 1 þ þ þO 9 : 6 7 8 725760n 241920n 576000n n
ð2:5Þ
From Lemma 1.1, we know that the speed of convergence of the sequence ðt n ÞnP1 is even higher as the value s satisfying (1.18). Thus, using Lemma 1.1, we have: (i) If a1 –
2369 , 302400
lim n5 t n ¼
n!1
then the rate of convergence of the sequence ðt n ÞnP1 is n5 , since
302400a1 þ 2369 – 0: 3628800
2369 (ii) If a1 ¼ 302400 , then from (2.5), we have
tn t nþ1 ¼
6157 1 þO 9 ; 8 1555200n n
and the rate of convergence of the sequence ðtn ÞnP1 is n7 , since
lim n7 t n ¼
n!1
6157 : 10886400
2369 We know that the fastest possible sequence ðt n ÞnP1 is obtained only for a1 ¼ 302400 .
Next, we define the sequence ðwn ÞnP1 by the relation
!nþ 2369 3 þa25 n 302400n pffiffiffiffiffiffiffiffiffinn 1 n! 2pn 1þ expðwn Þ: 1 e 12n2 10
ð2:6Þ
Using the same method from (2.1)–(2.5), we have that the fastest possible sequence ðwn ÞnP1 is obtained only for 6157 a2 ¼ 907200 . 14162903 Inductively, a3 ¼ 1437004800 ; , and the new approximation (1.12) is obtained. We can also obtain the approximations (1.13) and (1.14) by using the same method in the proof of (1.12). Here, we omitted the details. 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 P 0,
n X 1 1 B2j ln x x þ ln 2p þ ln Cðx þ 1Þ ¼ x þ þ ð1Þn Rn ðxÞ; 2 2 2jð2j 1Þx2j1 j¼1
ð3:1Þ
where Rn ðxÞ is completely monotonic on ð0; 1Þ; Bj is the jth Bernoulli number defined by the power series expansion
ex
1 1 X x xl x X x2l ¼ Bl ¼ 1 þ B2l : 2 l¼1 1 l! ð2lÞ! l¼0
ð3:2Þ
(B2lþ1 ¼ 0, for all l P 1, and the first few Bernoulli numbers are B1 ¼ 1=2; B2 ¼ 1=6; B4 ¼ 1=30; B6 ¼ 1=42). From (3.1), Mortici obtained the following inequalities in [12,13], for x > 0:
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
1 1 1 1 1 691 þ þ 12x 360x3 1260x5 1680x7 1188x9 360360x11 1 1 1 1 1 : < exp þ þ 12x 360x3 1260x5 1680x7 1188x9
exp
5
Cðx þ 1Þ < pffiffiffiffiffiffiffiffiffixx 2p x e ð3:3Þ
Next we give the proof of first inequalities in Theorem 1.2. For the right inequality (1.15) in Theorem 1.2, combining (3.3), we need to get
1 1 1 1 1 exp þ þ 12x 360x3 1260x5 1680x7 1188x9
0, for every x 2 ½2; 1Þ. Thus, f is strictly increasing on ½2; 1Þ, with f ð1Þ ¼ 0, so f ðxÞ < 0, for every x 2 ½2; 1Þ, then the right inequality (1.15) in Theorem 1.2 is obtained. For the left inequality (1.15) in Theorem 1.2, combining (3.3), we need to get
!xþ 2369 3 6157 5 907200x 302400x 1 1 1 1 1 1 691 : 1þ < exp þ þ 1 12x 360x3 1260x5 1680x7 1188x9 360360x11 12x2 10
ð3:7Þ
Inequality (3.7) is equivalent to gðxÞ > 0, where
gðxÞ ¼
1 2369 6157 x þ 302400x 3 907200x5
!
! 1 1 1 1 1 691 1 : ln 1 þ þ þ 1 12x 360x3 1260x5 1680x7 1188x9 360360x11 12x2 10
ð3:8Þ
We have
g 0 ðxÞ ¼
40 Gðx 2Þ ; 143 x7 ð120x2 1Þð40x2 þ 3Þð907200x6 þ 7107x2 6157Þ2
ð3:9Þ
where
GðxÞ ¼ 15994175570000x12 þ 2783860213680000x11 þ 30305400687156000x10 þ 197808515736720000x9 þ 861581579139338150x8 þ 2635038117227170400x7 þ 5791729104530007002x6 þ 9192725022193805624x5 þ 10413632818057286729x4 þ 8157265016456769992x3 þ 4148422041687612968x2 þ 1205307635021848160x þ 144918239064982466: g 0 ðxÞ < 0, for every x 2 ½2; 1Þ. Thus, g is strictly decreasing on ½2; 1Þ, with gð1Þ ¼ 0, so gðxÞ > 0, for every x 2 ½2; 1Þ, then the left inequality (1.15) in Theorem 1.2 is obtained. Since the proofs of inequalities (1.16) and (1.17) are the same to the one of inequality (1.15), we also omitted the details here. Remark 3.1. In fact, if we let
!xþ 2369 3 302400x 1 1 1 1 1 1 1 þ þ þ ; 1 3 5 7 9 2 12x 360x 1260x 1680x 1188x 12x 10
UðxÞ ¼ exp
6
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
VðxÞ ¼ exp
!xþ 2369 3 6157 5 907200x 302400x 1 1 1 1 1 691 1 1 þ þ þ : 1 3 5 7 9 11 2 12x 360x 1260x 1680x 1188x 360260x 12x 10
By some simulations, we obtain the following figure. From Fig. 1, it is easy to see that UðxÞ < 0 and VðxÞ > 0 uniformly on ½1:6; 1Þ. 4. Numerical computation In this section, we give Table 1 to demonstrate the superiority of our new series respectively. Mortici’s formula (1.9),
0 pffiffiffiffiffiffiffiffiffi B n n! 2pne @n þ
1n 1 12n
1 2369 10nþ 252 n
C A ¼ an :
ð4:1Þ
The new formula (1.12),
n!
!nþ 2369 3 302400n pffiffiffiffiffiffiffiffiffinn 1 1þ ¼ An : 2pn 1 2 e 12n 10
ð4:2Þ
The formula (1.7) as k = 6,
6 pffiffiffiffiffiffiffiffiffinn 1 1 1 11 79 1þ þ 2þ þ ¼ bn : 2pn e 2n 8n 240n3 1920n4 26880n5 1
n!
ð4:3Þ
The new formula (1.13),
n!
16 11 3 þ 13 4 pffiffiffiffiffiffiffiffiffinn 5760n 6720n 1 1 1 1þ þ 2þ ¼ Bn : 2pn 3 e 2n 8n 240n
ð4:4Þ
The formula (1.10),
pffiffiffiffiffiffiffin þ 1 2 2p e
nþ12
n!
0
1n12
B 1 B 24 B1 þ 41 n B @ n2 24014n
C C C C A
¼ cn :
41 xþ 1695131 nþ 2892960
Fig. 1. Simulations for UðxÞ and VðxÞ.
ð4:5Þ
7
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7 Table 1 Simulations for an ; An ; bn ; Bn ; cn and C n . ðan n!Þ=n!
n
ðAn n!Þ=n!
1:4815 10
2:2543 10
100
1:4816 1022
7:2392 1023
2:3320 1020
500
1:1575 10
24
25
22
2500
1:8965 1027
15
7:2350 10
16
5:6557 10
9:2663 1028
14
3:6505 10
ðn! cn Þ=n!
ðBn n!Þ=n!
ðn! bn Þ=n!
50
1:4969 1024
15
ðC n n!Þ=n!
2:6686 10
14
4:9363 10
8:3704 1015
2:0169 1021
5:0542 1020
7:5463 1021
23
3:0948 10
22
7:9075 10
1:1720 1022
1:2537 1025
3:2414 1024
4:7828 1025
The new formula (1.14),
n!
!n12þ 7 2 52531 3 1 120n 1209600n pffiffiffiffiffiffiffin þ 1nþ2 1 2 1 þ 2 2441 ¼ Cn: 2p e n 240
ð4:6Þ
Combining Theorem 1.2, we have Table 1. From Table 1, it is easy to see that our new formula (4.2), (4.4) and (4.6) are more accurate than formula (4.1), (4.3) and (4.5) respectively. Acknowledgement We thank the referees for careful reading of our paper and for helpful and valuable comments and suggestions. The comments by the referees helped the author improve the exposition of this paper significantly. The research of the first author is supported by the National Natural Science Foundation of China (Grant Nos. 11101061 and 11371077). The second author is supported by the National Natural Science Foundation of China (Grant Nos. 11371077 and 61175041). References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nation Bureau of Standards, Applied Mathematical Series, 55, 9th Printing, Dover, New York, 1972. [2] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comput. 66 (217) (1997) 373–389. [3] W. Burnside, A rapidly convergent series for log N!, Messenger Math 46 (1917) 157–159. [4] R.W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40–42. [5] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publ. H., Springer, New Delhi, Berlin, 1988. Intr. by G.E. Andrews. [6] http://www.rskey.org/gamma.htm [7] G. Nemes, New asymptotic expansion for the Gamma function, Arch. Math. 95 (2010) 161–169. [8] C. Mortici, Very accurate estimates of the polygamma functions, Asymptot. Anal. 68 (3) (2010) 125–134. [9] C. Mortici, A quicker convergence toward the gamma constant with the logarithm term involving the constant e, Carpath. J. Math. 26 (1) (2010) 86–91. [10] C. Mortici, On new sequences converging towards the Euler–Mascheroni constant, Comput. Math. Appl. 59 (8) (2010) 2610–2614. [11] C. Mortici, Product approximations via asymptotic integration, Am. Math. Monthly 117 (5) (2010) 434–441. [12] C. Mortici, A new Stirling series as continued fraction, Numer. Algorithms 56 (1) (2011) 17–26. [13] C. Mortici, A continued fraction approximation of the gamma function, J. Math. Anal. Appl. 402 (2013) 405–410. [14] D. Lu, X. Wang, A generated approximation related to Gosper’s formula and Ramanujan’s formula, J. Math. Anal. Appl. 406 (2013) 287–292. [15] D. Lu, L. Song, C. Ma, A generated approximation of the gamma function related to Windschitl’s formula, J. Number Theory 140 (2014) 215–225. [16] D. Lu, X. Wang, A new asymptotic expansion and some inequalities for the gamma function, J. Number Theory 140 (2014) 314–323. [17] C.-P. Chen, Asymptotic expansions of the gamma function related to Windschitl’s formula, Appl. Math. Comput. 245 (2014) 174–180.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Some new asymptotic approximations of the gamma function based on Nemes’ formula, Ramanujan’s formula and Burnside’s formula Dawei Lu, Lixin Song ⇑, Congxu Ma School 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 In this paper, we construct some new approximations of the gamma function based on Nemes’ formula, Ramanujan’s formula and Burnside’s formula. Using these approximations, some inequalities are established. Finally, for demonstrating the superiority of our new approximation over Mortici’s formula and other classical ones, some numerical computations are also given. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Gamma function Nemes’ formula Ramanujan’s formula Burnside’s formula Asymptotic approximations Rate of convergence
1. Introduction We often need to deal with big factorials in many areas of mathematics and science in general. Undoubtedly, one of the most known formulas for approximation of the factorial function is Stirling’s formula [1],
n!
pffiffiffiffiffiffiffiffiffinn : 2pn e
ð1:1Þ
The next step in this direction is to define more accurate approximations for the factorial function. A more accurate approximation than formula (1.1) is Burnside’s formula [3],
pffiffiffiffiffiffiffin þ 1nþ2 2 : 2p e 1
n!
ð1:2Þ
There are two approximations which are better than (1.2), Gosper’s formula [4],
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1 nn n! 2p n þ ; 6 e
ð1:3Þ
and Ramanujan’s formula [5],
n!
pffiffiffiffiffiffiffinn 2p e
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 1 1 1 n3 þ n2 þ n þ : 2 8 240
Windschitl’s formula [6],
⇑ Corresponding author. E-mail addresses: [email protected] (D. Lu), [email protected] (L. Song), [email protected] (C. Ma). http://dx.doi.org/10.1016/j.amc.2014.12.077 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð1:4Þ
2
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
n!
n=2 pffiffiffiffiffiffiffiffiffinn 1 n sinh ; 2pn e n
ð1:5Þ
is more exact. Nemes’ formula [7],
!n pffiffiffiffiffiffiffiffiffinn 1 n! 2pn 1þ ; 1 e 12n2 10
ð1:6Þ
has the same number of exact digits as (1.5) but is much simpler. Recently, many researchers presented some more accurate approximations for the factorial function and its extension gamma function. Lu [14] presented a polynomial approximation for the factorial function as follows,
n!
1=k pffiffiffiffiffiffiffiffiffinn c1 c2 c3 1 þ þ 2 þ 3 þ ; 2pn e n n n
c1 ¼
k ; 12
ð1:7Þ
where 2
c2 ¼
k ; 288
2
c3 ¼
kð5k 144Þ ; 51840
Based on the works of Windschitl and Mortici, Lu [15] provided a formula which is an extension of Windschitl’s formula as follows,
n!
n=2 pffiffiffiffiffiffiffiffiffinn 1 a7 a9 a11 n sinh þ 7 þ 9 þ 11 þ ; 2pn e n n n n
a7 ¼
1 ; 810
ð1:8Þ
where
a9 ¼
67 ; 42525
a11 ¼
19 ; 8505
In addition, some authors paid attention to giving increasingly better approximations for the gamma function by using the continued fractions. For example, Mortici [13] provided a new continued fraction approximation starting from Nemes’ formula (1.4) as follows,
1x
0
Cðx þ 1Þ
B pffiffiffiffiffiffiffiffiffi B 1 B 2pxex Bx þ B 12x 10xþ 1 @ xþ
a b
c xþ xþ d . xþ . .
C C C C; C A
ð1:9Þ
where
a¼
2369 ; 252
b¼
2117009 ; 1193976
c¼
393032191511 ; 1324011300744
d¼
33265896164277124002451 ; 14278024104089641878840
Using the continued fractions, Lu [16] provided an asymptotic expansion for the factorial function starting from Burnside’s formula (1.2) as follows,
0 1 xþ 2B pffiffiffiffiffiffiffi x þ 1 a1 B 2 Cðx þ 1Þ 2p B1 þ 2 @ e x þ a2ax3 x xþ
a x xþ 4 . xþ . .
1x12 C C C A
;
ð1:10Þ
where
a1 ¼
1 ; 24
a2 ¼
41 ; 240
a3 ¼
14 ; 41
a4 ¼
1695131 ; 2892960
Very recently, using a different method, Chen [17] gave a new approximation related to Windschitl’s formula,
Cðx þ 1Þ
xþ 1 þ 191 þ 25127 þ pffiffiffiffiffiffiffiffiffixx 1 2 270x3 56700x5 5103000x7 x sinh : 2px e x
ð1:11Þ
It is their works that motivate our study. In this paper, based on Nemes’ formula (1.6), Ramanujan’s formula (1.4), the extension of Burnside’s formula (1.10) and the works of Chen [17], using the same method in Mortici [13], we introduce some new approximations as follows,
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
3
Theorem 1.1. For the factorial function, based on (1.6), we have
n!
!nþ 2369 3 6157 5 þ 14162903 7 þ 907200n 1437004800n 302400n pffiffiffiffiffiffiffiffiffinn 1 2pn 1þ : 1 2 e 12n 10
ð1:12Þ
Based on (1.4), we have
n!
16 11 3 þ 13 4 7 5 þ pffiffiffiffiffiffiffiffiffinn 115200n 5760n 6720n 1 1 1 1þ þ 2þ : 2pn e 2n 8n 240n3
ð1:13Þ
Based on (1.10), we have
!n12þ 7 2 52531 3 þ 6763 4 þ 1 1 120n 1209600n 2419200n pffiffiffiffiffiffiffin þ 1nþ2 24 2 n! 2p 1 þ 2 41 : e n 240
ð1:14Þ
Next, using Theorem 1.1, we provide some inequalities for the gamma function. Theorem 1.2. (i) For every x > 2, it holds:
!xþ 2369 3 6157 5 907200x 302400x 1 Cðx þ 1Þ 1þ < pffiffiffiffiffiffiffiffiffixx < 1 2 12x 10 2px e
!xþ 2369 3 302400x 1 1þ : 1 2 12x 10
ð1:15Þ
(ii) For every x > 2, it holds:
16 11 3 16 11 3 þ 13 4 5760x 5760x 6720x 1 1 1 Cðx þ 1Þ 1 1 1 p ffiffiffiffiffiffiffiffi ffi 1þ þ 2þ < < 1 þ þ þ : 2 3 x x 2x 8x 240x3 2x 8x 240x 2px e
ð1:16Þ
(iii) For every x > 2, it holds:
!x12þ 7 2 1 120x 24 Cðx þ 1Þ 1 þ 2 41 < < pffiffiffiffiffiffiffixþ1xþ12 x 240 2p e 2
!x12þ 7 2 52531 3 1 120x 1209600x 24 1 þ 2 41 : x 240
ð1:17Þ
To obtain Theorem 1.1, we need the following lemma which was used in [8–10,13] and is very useful for constructing asymptotic expansions. Lemma 1.1. If ðxn ÞnP1 is convergent to zero and there exists the limit
lim ns ðxn xnþ1 Þ ¼ l 2 ½1; þ1;
n!1
ð1:18Þ
with s > 1, then
lim ns1 xn ¼
n!1
l : s1
ð1:19Þ
Lemma 1.1 was first proved by Mortici in [11]. From Lemma 1.1, we can see that the speed of convergence of the sequence ðxn ÞnP1 increases together with the value s satisfying (1.18). The rest of paper is arranged as follows. In Section 2, the proof of Theorem 1.1 is given. In Section 3, we provide the proof of Theorem 1.2. In Section 4, we give some numerical computations to demonstrate the superiority of our new series respectively. 2. Proof of Theorem 1.1 Based on the argument of Theorem 2.1 in [12] or Theorem 5 in [13], we need to find the value a1 2 R which produces the most accurate approximation of the form
!nþa13 n pffiffiffiffiffiffiffiffiffinn 1 n! 2pn 1þ : 1 e 12n2 10
ð2:1Þ
To measure the accuracy of this approximation, a method is to define a sequence ðt n ÞnP1 by the relations
!nþa13 n pffiffiffiffiffiffiffiffiffinn 1 n! 2pn 1þ expðt n Þ; 1 2 e 12n 10
ð2:2Þ
4
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
and to say that an approximation (2.1) is better if t n converges to zero faster. From (2.2), we have
! 1 1 a1 1 : ln n þ n n þ 3 ln 1 þ tn ¼ ln n! ln 2p n þ 1 2 2 n 12n2 10
ð2:3Þ
Thus,
tn t nþ1
! 1 1 a1 1 ln 1 þ n þ 3 ln 1 þ ¼ 1 þ n þ 1 2 n n 12n2 10 ! ! a1 1 ln 1 þ : þ nþ1þ 1 12ðn þ 1Þ2 10 ðn þ 1Þ3
ð2:4Þ
Developing the power series in 1=n, we have
tn t nþ1 ¼
302400a1 þ 2369 302400a1 þ 2369 1668800a1 þ 10793 1 þ þ þO 9 : 6 7 8 725760n 241920n 576000n n
ð2:5Þ
From Lemma 1.1, we know that the speed of convergence of the sequence ðt n ÞnP1 is even higher as the value s satisfying (1.18). Thus, using Lemma 1.1, we have: (i) If a1 –
2369 , 302400
lim n5 t n ¼
n!1
then the rate of convergence of the sequence ðt n ÞnP1 is n5 , since
302400a1 þ 2369 – 0: 3628800
2369 (ii) If a1 ¼ 302400 , then from (2.5), we have
tn t nþ1 ¼
6157 1 þO 9 ; 8 1555200n n
and the rate of convergence of the sequence ðtn ÞnP1 is n7 , since
lim n7 t n ¼
n!1
6157 : 10886400
2369 We know that the fastest possible sequence ðt n ÞnP1 is obtained only for a1 ¼ 302400 .
Next, we define the sequence ðwn ÞnP1 by the relation
!nþ 2369 3 þa25 n 302400n pffiffiffiffiffiffiffiffiffinn 1 n! 2pn 1þ expðwn Þ: 1 e 12n2 10
ð2:6Þ
Using the same method from (2.1)–(2.5), we have that the fastest possible sequence ðwn ÞnP1 is obtained only for 6157 a2 ¼ 907200 . 14162903 Inductively, a3 ¼ 1437004800 ; , and the new approximation (1.12) is obtained. We can also obtain the approximations (1.13) and (1.14) by using the same method in the proof of (1.12). Here, we omitted the details. 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 P 0,
n X 1 1 B2j ln x x þ ln 2p þ ln Cðx þ 1Þ ¼ x þ þ ð1Þn Rn ðxÞ; 2 2 2jð2j 1Þx2j1 j¼1
ð3:1Þ
where Rn ðxÞ is completely monotonic on ð0; 1Þ; Bj is the jth Bernoulli number defined by the power series expansion
ex
1 1 X x xl x X x2l ¼ Bl ¼ 1 þ B2l : 2 l¼1 1 l! ð2lÞ! l¼0
ð3:2Þ
(B2lþ1 ¼ 0, for all l P 1, and the first few Bernoulli numbers are B1 ¼ 1=2; B2 ¼ 1=6; B4 ¼ 1=30; B6 ¼ 1=42). From (3.1), Mortici obtained the following inequalities in [12,13], for x > 0:
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
1 1 1 1 1 691 þ þ 12x 360x3 1260x5 1680x7 1188x9 360360x11 1 1 1 1 1 : < exp þ þ 12x 360x3 1260x5 1680x7 1188x9
exp
5
Cðx þ 1Þ < pffiffiffiffiffiffiffiffiffixx 2p x e ð3:3Þ
Next we give the proof of first inequalities in Theorem 1.2. For the right inequality (1.15) in Theorem 1.2, combining (3.3), we need to get
1 1 1 1 1 exp þ þ 12x 360x3 1260x5 1680x7 1188x9
0, for every x 2 ½2; 1Þ. Thus, f is strictly increasing on ½2; 1Þ, with f ð1Þ ¼ 0, so f ðxÞ < 0, for every x 2 ½2; 1Þ, then the right inequality (1.15) in Theorem 1.2 is obtained. For the left inequality (1.15) in Theorem 1.2, combining (3.3), we need to get
!xþ 2369 3 6157 5 907200x 302400x 1 1 1 1 1 1 691 : 1þ < exp þ þ 1 12x 360x3 1260x5 1680x7 1188x9 360360x11 12x2 10
ð3:7Þ
Inequality (3.7) is equivalent to gðxÞ > 0, where
gðxÞ ¼
1 2369 6157 x þ 302400x 3 907200x5
!
! 1 1 1 1 1 691 1 : ln 1 þ þ þ 1 12x 360x3 1260x5 1680x7 1188x9 360360x11 12x2 10
ð3:8Þ
We have
g 0 ðxÞ ¼
40 Gðx 2Þ ; 143 x7 ð120x2 1Þð40x2 þ 3Þð907200x6 þ 7107x2 6157Þ2
ð3:9Þ
where
GðxÞ ¼ 15994175570000x12 þ 2783860213680000x11 þ 30305400687156000x10 þ 197808515736720000x9 þ 861581579139338150x8 þ 2635038117227170400x7 þ 5791729104530007002x6 þ 9192725022193805624x5 þ 10413632818057286729x4 þ 8157265016456769992x3 þ 4148422041687612968x2 þ 1205307635021848160x þ 144918239064982466: g 0 ðxÞ < 0, for every x 2 ½2; 1Þ. Thus, g is strictly decreasing on ½2; 1Þ, with gð1Þ ¼ 0, so gðxÞ > 0, for every x 2 ½2; 1Þ, then the left inequality (1.15) in Theorem 1.2 is obtained. Since the proofs of inequalities (1.16) and (1.17) are the same to the one of inequality (1.15), we also omitted the details here. Remark 3.1. In fact, if we let
!xþ 2369 3 302400x 1 1 1 1 1 1 1 þ þ þ ; 1 3 5 7 9 2 12x 360x 1260x 1680x 1188x 12x 10
UðxÞ ¼ exp
6
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7
VðxÞ ¼ exp
!xþ 2369 3 6157 5 907200x 302400x 1 1 1 1 1 691 1 1 þ þ þ : 1 3 5 7 9 11 2 12x 360x 1260x 1680x 1188x 360260x 12x 10
By some simulations, we obtain the following figure. From Fig. 1, it is easy to see that UðxÞ < 0 and VðxÞ > 0 uniformly on ½1:6; 1Þ. 4. Numerical computation In this section, we give Table 1 to demonstrate the superiority of our new series respectively. Mortici’s formula (1.9),
0 pffiffiffiffiffiffiffiffiffi B n n! 2pne @n þ
1n 1 12n
1 2369 10nþ 252 n
C A ¼ an :
ð4:1Þ
The new formula (1.12),
n!
!nþ 2369 3 302400n pffiffiffiffiffiffiffiffiffinn 1 1þ ¼ An : 2pn 1 2 e 12n 10
ð4:2Þ
The formula (1.7) as k = 6,
6 pffiffiffiffiffiffiffiffiffinn 1 1 1 11 79 1þ þ 2þ þ ¼ bn : 2pn e 2n 8n 240n3 1920n4 26880n5 1
n!
ð4:3Þ
The new formula (1.13),
n!
16 11 3 þ 13 4 pffiffiffiffiffiffiffiffiffinn 5760n 6720n 1 1 1 1þ þ 2þ ¼ Bn : 2pn 3 e 2n 8n 240n
ð4:4Þ
The formula (1.10),
pffiffiffiffiffiffiffin þ 1 2 2p e
nþ12
n!
0
1n12
B 1 B 24 B1 þ 41 n B @ n2 24014n
C C C C A
¼ cn :
41 xþ 1695131 nþ 2892960
Fig. 1. Simulations for UðxÞ and VðxÞ.
ð4:5Þ
7
D. Lu et al. / Applied Mathematics and Computation 253 (2015) 1–7 Table 1 Simulations for an ; An ; bn ; Bn ; cn and C n . ðan n!Þ=n!
n
ðAn n!Þ=n!
1:4815 10
2:2543 10
100
1:4816 1022
7:2392 1023
2:3320 1020
500
1:1575 10
24
25
22
2500
1:8965 1027
15
7:2350 10
16
5:6557 10
9:2663 1028
14
3:6505 10
ðn! cn Þ=n!
ðBn n!Þ=n!
ðn! bn Þ=n!
50
1:4969 1024
15
ðC n n!Þ=n!
2:6686 10
14
4:9363 10
8:3704 1015
2:0169 1021
5:0542 1020
7:5463 1021
23
3:0948 10
22
7:9075 10
1:1720 1022
1:2537 1025
3:2414 1024
4:7828 1025
The new formula (1.14),
n!
!n12þ 7 2 52531 3 1 120n 1209600n pffiffiffiffiffiffiffin þ 1nþ2 1 2 1 þ 2 2441 ¼ Cn: 2p e n 240
ð4:6Þ
Combining Theorem 1.2, we have Table 1. From Table 1, it is easy to see that our new formula (4.2), (4.4) and (4.6) are more accurate than formula (4.1), (4.3) and (4.5) respectively. Acknowledgement We thank the referees for careful reading of our paper and for helpful and valuable comments and suggestions. The comments by the referees helped the author improve the exposition of this paper significantly. The research of the first author is supported by the National Natural Science Foundation of China (Grant Nos. 11101061 and 11371077). The second author is supported by the National Natural Science Foundation of China (Grant Nos. 11371077 and 61175041). References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Nation Bureau of Standards, Applied Mathematical Series, 55, 9th Printing, Dover, New York, 1972. [2] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comput. 66 (217) (1997) 373–389. [3] W. Burnside, A rapidly convergent series for log N!, Messenger Math 46 (1917) 157–159. [4] R.W. Gosper, Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA 75 (1978) 40–42. [5] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publ. H., Springer, New Delhi, Berlin, 1988. Intr. by G.E. Andrews. [6] http://www.rskey.org/gamma.htm [7] G. Nemes, New asymptotic expansion for the Gamma function, Arch. Math. 95 (2010) 161–169. [8] C. Mortici, Very accurate estimates of the polygamma functions, Asymptot. Anal. 68 (3) (2010) 125–134. [9] C. Mortici, A quicker convergence toward the gamma constant with the logarithm term involving the constant e, Carpath. J. Math. 26 (1) (2010) 86–91. [10] C. Mortici, On new sequences converging towards the Euler–Mascheroni constant, Comput. Math. Appl. 59 (8) (2010) 2610–2614. [11] C. Mortici, Product approximations via asymptotic integration, Am. Math. Monthly 117 (5) (2010) 434–441. [12] C. Mortici, A new Stirling series as continued fraction, Numer. Algorithms 56 (1) (2011) 17–26. [13] C. Mortici, A continued fraction approximation of the gamma function, J. Math. Anal. Appl. 402 (2013) 405–410. [14] D. Lu, X. Wang, A generated approximation related to Gosper’s formula and Ramanujan’s formula, J. Math. Anal. Appl. 406 (2013) 287–292. [15] D. Lu, L. Song, C. Ma, A generated approximation of the gamma function related to Windschitl’s formula, J. Number Theory 140 (2014) 215–225. [16] D. Lu, X. Wang, A new asymptotic expansion and some inequalities for the gamma function, J. Number Theory 140 (2014) 314–323. [17] C.-P. Chen, Asymptotic expansions of the gamma function related to Windschitl’s formula, Appl. Math. Comput. 245 (2014) 174–180.
