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Computers and Mathematics with Applications 60 (2010) 786–791
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Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
The asymptotic series of the generalized Stirling formula Cristinel Mortici ∗ Valahia University of Târgovişte, Department of Mathematics, Bd. Unirii 18, 130082 Târgovişte, Romania article info a b s t r a c t
Article history: The aim of this paper is to construct some asymptotic expansions which produce Received 11 January 2010 increasingly accurate approximations of the gamma function. Received in revised form 20 May 2010 © 2010 Elsevier Ltd. All rights reserved. Accepted 20 May 2010 keywords: Approximations Completely monotonicity Asymptotic series Gamma function Stirling formula Burnside formula
1. Introduction
In recent years, several authors have concentrating on establishing new approximations for bounding the gamma function and related functions. We refer the reader, for example, to [1–17] and all references therein. These new increasingly better estimates are of great importance in many branches of science such as probabilities, applied statistics, statistical physics, or in optimization theory for improving the analysis of some algorithms. Perhaps, one of the most known and most used formulae for approximation of the gamma function is the following
√ x+ 1 x 2 Γ (x + 1) ≈ 2πe , (1.1) e now known as Stirling’s formula. A slightly more accurate approximation was proposed in [4]
r x+ 1 2π x + 1 2 Γ (x + 1) ≈ (1.2) e e as an intermediate result for establishing some monotonicity properties of some functions involving the gamma function. However, Burnside’s approximation [18]
!x+ 1 √ x + 1 2 Γ (x + 1) ≈ 2π 2 (1.3) e
∗ Tel.: +40 722727627; fax: +40 0245213382. E-mail addresses: [email protected], [email protected]. URL: http://www.cristinelmortici.ro.
0898-1221/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.05.025 C. Mortici / Computers and Mathematics with Applications 60 (2010) 786–791 787 is much better. It is introduced in [4] the following class of approximations
+ 1 √ x 2 − x + p Γ (x + 1) ≈ e p 2πe (p ∈ [0, 1]), (1.4) e which incorporates (1.2) (p = 1) and also the famous estimates due to Stirling (p = 0) and Burnside (p = 1/2). This is the reason which allows us to call (1.4) the generalized Stirling formula. Furthermore, it is proven in [4] that the most accurate √ approximations (1.4) are obtained for p = 3 ± 3 /6. This problem of extending known results about the Stirling formula to the entire class of approximations (1.4) was continued in the recent paper [5] where the following integral representation associated to the Stirling formula due to Liu [19] ! √ n n Z ∞ 1 − {t} Γ (n + 1) = 2πn exp 2 dt e n t was extended for every p ∈ [0, 1] to
n+ 1 3 ! ! √ 2 Z ∞ − n + p − p − {t} p 1 Γ (n + 1) = e p 2πe exp 2 + − dt . e n t + p p{t} + [t] t
([t] denotes the largest integer less than or equal to t and {t} = t − [t]). The Stirling formula (1.1) is in fact the first approximation of the following asymptotic series
∞ ! √ x+ 1 x 2 X Bk+1 Γ (x + 1) ∼ 2πe exp . (1.5) e k k + 1 xk k=1 ( )
(Bj denotes the jth Bernoulli number). See, e.g., [20, p. 257 and p. 804]. One of our main results is the asymptotic expansion associated to the generalized Stirling formula (1.4).
Theorem 1. The following asymptotic expansion holds as x → ∞
+ 1 ∞ ! √ x 2 − x + p X ap(k) Γ (x + 1) ∼ e p 2πe exp (p ∈ [0, 1]) , (1.6) e xk k=1 where for every integer k ≥ 1, 1 k k 1 1 ap(k) = Bk+1 + (−1) p − p − k . k (k + 1) 2 2
It can be easily seen that case p = 0 is the Stirling series (1.5). Cases p = 1/2 and p = 1 related to the approximation formulas (1.3) and (1.2) respectively, are gathered in the following corollaries.
Corollary 1. The following Burnside asymptotic series holds as x → ∞
!x+ 1 ! √ + 1 2 ∞ − k x 2 X ( 1) 1 Γ (x + 1) ∼ 2π exp Bk+1 + , (1.7) e 2k+1 k k + 1 xk k=1 ( ) namely
!x+ 1 √ x + 1 2 Γ (x + 1) ∼ 2π 2 e 1 1 23 1 11 1 143 × exp − + − + + + − ··· . 24x 48x2 2880x3 640x4 40320x5 5376x6 215040x7
Corollary 2. The following asymptotic series holds as x → ∞
r x+ 1 ∞ ! + 2 − 2π x 1 X k k 1 1 Γ (x + 1) ∼ exp Bk+1 − (−1) , (1.8) e e 2 k k + 1 xk k=1 ( ) 788 C. Mortici / Computers and Mathematics with Applications 60 (2010) 786–791 namely
r x+ 1 2π x + 1 2 1 1 29 3 17 5 89 Γ (x + 1) ∼ · exp − + − + − + ··· . e e 12x 12x2 360x3 40x4 252x5 84x6 1680x7
Asymptotic series like (1.6) and their particular forms (1.7)–(1.8) may not, and often do not converge, but in a truncated form of only a few terms, they provide approximations to any desired accuracy, as in the estimate
+ 1 ( ) √ x 2 m − x + p X ap (k) Γ (x + 1) ≈ e p 2πe exp . e xk k=1 The next step in this approximation problem is to establish sharp inequalities involving the first m terms of the respective asymptotic series. We give here some particular examples which allow us to develop a general method for discovering such increasingly accurate bounds. The following sharp bounds are related to the asymptotic series (1.7)–(1.8) and also to a general case of the parameter p from (1.6).
Theorem 2. For every x ≥ 0, we have
!x+ 1 !x+ 1 √ x + 1 2 √ x + 1 2 2π 2 exp a(x) < Γ (x + 1) < 2π 2 exp b(x), e e where 1 1 23 1 11 1 143 a(x) = − + − + + + − 24x 48x2 2880x3 640x4 40320x5 5376x6 215040x7 and b(x) = a(x) + 143 . 215 040x7
Theorem 3. For every x ≥ 1, we have
r x+ 1 r x+ 1 2π x + 1 2 2π x + 1 2 exp c(x) < Γ (x + 1) < exp d(x), e e e e where 1 1 29 3 17 5 c(x) = − + − + − 12x 12x2 360x3 40x4 252x5 84x6 and d(x) = c(x) + 89 . 1680x7
∈ 3 5 ∈ ∞ Theorem 4. For every p 5 , 8 and x (0, ), we have
( 3 ) ( 5 ) X ap(k) Γ (x + 1) X ap (k) exp < < exp . k √ 1 k x x+p x+ x k=1 −p 2 k=1 e 2πe e
Inspired by the Stirling series (1.5), Alzer [1, Theorem 8] proved that for every n ≥ 0, the functions
2n Γ (x + 1) X B2j F (x) = ln − n √ 1 2j−1 x x+ 2j 2j − 1 x 2 j=1 ( ) 2πe e and
2n+1 Γ (x + 1) X B2j G (x) = − ln + n √ 1 2j−1 x x+ 2j 2j − 1 x 2 j=1 ( ) 2πe e are completely monotonic on (0, ∞) and as a direct consequence of Fn > 0 and Gn > 0, he deduced
2n ! 2n+1 ! X B2j Γ (x + 1) X B2j exp < < exp . (1.9) 2j−1 √ 1 2j−1 2j 2j − 1 x x x+ 2j 2j − 1 x j=1 ( ) 2 j=1 ( ) 2πe e C. Mortici / Computers and Mathematics with Applications 60 (2010) 786–791 789
In order to extend Alzer’s result to the case of generalized Stirling formula, we denote
Γ (x + 1) exp λp(x) = (p ∈ [0, 1]). (1.10) √ x+ 1 −p x+p 2 e 2πe e
Remark that the function λ0 produces the Stirling approximation. Straightforward computations lead us to
1 p λp(x) = λ0(x) + p − x + ln 1 + . (1.11) 2 x
Using the notations (1.10), the inequalities (1.9) can be equivalently written as
2n 2n+1 X B2j X B2j < λ0(x) < (1.12) 2j 2j − 1 x2j−1 2j 2j − 1 x2j−1 j=1 ( ) j=1 ( ) and we extend them to the following.
Theorem 5. For every p ∈ [0, 1] , x ∈ (0, ∞) and positive integer n, we have
2n 1 p X B2j p − x + ln 1 + + 2 x 2j 2j − 1 x2j−1 j=1 ( ) 2n+1 1 p X B2j < λp(x) < p − x + ln 1 + + . (1.13) 2 x 2j 2j − 1 x2j−1 j=1 ( )
Moreover, for every p ∈ [0, 1], the functions
2n Γ (x + 1) 1 p X B2j F (x) = ln − p + x + ln 1 + − n,p √ 1 2j−1 x+p x+ 2 x 2j 2j − 1 x −p 2 j=1 ( ) e 2πe e and 2n+1 Γ (x + 1) 1 p X B2j G (x) = − ln + p − x + ln 1 + + n,p √ 1 2j−1 x+p x+ 2 x 2j 2j − 1 x −p 2 j=1 ( ) e 2πe e are completely monotonic on (0, ∞).
Alzer’s inequality (1.9) is obtained from Theorem 5 in case p = 0, while Fn = Fn,0 and Gn = Gn,0. The proof of Theorem 5 follows immediately using the complete monotonicity of functions Fn and Gn, inequalities (1.9) and the connecting relation (1.11), but as we can see in the next section, written in this form, it is very useful in practical problems.
2. The proofs
First note that easy computations lead us to the following representation in power series ∞ 1 p X yp(k) up(x) = p − x + ln 1 + = , (2.1) 2 x xk k=1 where for every integer k ≥ 1, − − k−1 k (2p 1) k 1 yp(k) = (−1) p . 2k (k + 1)
Using relation (1.11) in the form λp(x) = λ0(x) + up(x), the asymptotic series (1.6) follows by adding the Stirling series (1.5) with the power series expansion of up to obtain
Bk+1 ap(k) = + yp (k) . k (k + 1) In this way, Theorem 1 is proved. 790 C. Mortici / Computers and Mathematics with Applications 60 (2010) 786–791
Proof of Theorem 2. In terms of (1.10), we have to prove
a(x) < λ1/2(x) < b(x). By Theorem 5, we have
4 3 X B2j X B2j u1/2(x) + < λ1/2(x) < u1/2(x) + , 2j 2j − 1 x2j−1 2j 2j − 1 x2j−1 j=1 ( ) j=1 ( ) so it suffices to show that f > 0 and g < 0, where
4 X B2j f (x) = u1/2(x) + − a(x) 2j 2j − 1 x2j−1 j=1 ( )
3 X B2j g(x) = u1/2(x) + − b(x). 2j 2j − 1 x2j−1 j=1 ( ) But f is strictly convex, g is strictly concave, since
00 1 00 1 f (x) = , g (x) = − , 256x9 (2x + 1) 128x8 (2x + 1) with f (∞) = g (∞) = 0, so f > 0 and g < 0 and the theorem is proved. Proof of Theorem 3. In terms of (1.10), we have to prove
c(x) < λ1(x) < d(x). By Theorem 5, we have
4 5 X B2j X B2j u1(x) + < λ1(x) < u1(x) + , 2j 2j − 1 x2j−1 2j 2j − 1 x2j−1 j=1 ( ) j=1 ( ) so it suffices to show that h > 0 and j < 0, where
4 X B2j h(x) = u1(x) + − c(x) 2j 2j − 1 x2j−1 j=1 ( )
5 X B2j j(x) = u1(x) + − d(x). 2j 2j − 1 x2j−1 j=1 ( ) But h is strictly convex, and j is strictly concave on [1, ∞), since
2 3 2 00 89x + 73x − 1 00 231x + 193x − 10x − 5 h (x) = , j (x) = − , 30x9 (x + 1)2 66x11 (x + 1)2 with f (∞) = g (∞) = 0, so f > 0 and g < 0 and the theorem is proved. In order to prove Theorem 4, we need the following
∈ 3 5 ∈ ∞ Lemma 1. For every p 5 , 8 and x (0, ), we have
3 5 X yp(k) X yp(k) < up(x) < . xk xk k=1 k=1
Proof. Let us denote m X yp(k) fm,p(x) = up(x) − (m ≥ 1). xk k=1 ∈ 3 5 For every p 5 , 8 , we have p4 (5 − 8p) 2p (2 − 3p) 00 = + f3,p(x) x > 0 2x5 (x + p)2 5 − 8p C. Mortici / Computers and Mathematics with Applications 60 (2010) 786–791 791 and p6 (12p − 7) 2p (5p − 3) 00 = − + f5,p(x) x < 0. 2x7 (p + x)2 12p − 7
Now f3,p is strictly convex, f5,p is strictly concave, with f3,p (∞) = f5,p (∞) = 0, so f3,p > 0 and f5,p < 0 and the conclusion follows. Proof of Theorem 4. By lemma,
3 5 X yp(k) X yp(k) < up(x) < , xk xk k=1 k=1 and from (1.12) with n = 1, we get
3 5 X Bk+1 X Bk+1 < λ0 (x) < . k k + 1 xk k k + 1 xk k=1 ( ) k=1 ( ) By addition, we obtain 3 5 X Bk+1 1 X Bk+1 1 + yp (k) < λ0(x) + up (x) < + yp(k) , k k + 1 xk k k + 1 xk k=1 ( ) k=1 ( ) or
3 5 X ap(k) X ap(k) < λp (x) < , xk xk k=1 k=1 which is the conclusion.
Finally it is to be noticed that for arbitrary given p ∈ [0, 1], sharp bounds for λp(x) of any desired accuracy can be constructed. The proof of such results can be reduced to the variation of some strictly convex (concave) functions that tend to zero, as the argument approaches infinity.
Acknowledgement
The computations in this paper were made using Maple software.
References
[1] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp. 66 (217) (1997) 373–389. [2] V. Berinde, A new generalization of Euler’s constant, Creat. Math. 18 (2) (2009) 123–128. [3] C. Mortici, Product approximation via asymptotic integration, Amer. Math. Monthly 117 (5) (2010) 434–441. [4] C. Mortici, An ultimate extremely accurate formula for approximation of the factorial function, Arch. Math. (Basel) 93 (1) (2009) 37–45. [5] C. Mortici, A class of integral approximations for the factorial function, Comput. Math. Appl. 59 (6) (2010) 2053–2058. [6] C. Mortici, Sharp inequalities related to Gosper’s formula, C. R. Math. Acad. Sci. Paris 348 (3–4) (2010) 137–140. [7] C. Mortici, Best estimates of the generalized Stirling formula, Appl. Math. Comput. 215 (11) (2010) 4044–4048. [8] C. Mortici, Improved convergence towards generalized Euler–Mascheroni constant, Appl. Math. Comput. 215 (2010) 3443–3448. [9] C. Mortici, New approximations of the gamma function in terms of the digamma function, Appl. Math. Lett. 23 (1) (2010) 97–100. [10] C. Mortici, Very accurate estimates of the polygamma functions, Asymptot. Anal. (2010) (in press). [11] C. Mortici, Optimizing the rate of convergence in some new classes of sequences convergent to Euler’s constant, Anal. Appl. (Singap.) 8 (1) (2010) 1–9. [12] C. Mortici, New sharp bounds for gamma and digamma functions, An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă Mat. Tom LVI (2) (2010) (in press). [13] C. Mortici, Complete monotonic functions associated with gamma function and applications, Carpathian J. Math. 25 (2) (2009) 186–191. [14] C. Mortici, Sharp inequalities and complete monotonicity for the Wallis ratio, Bull. Belg. Math. Soc. Simon Stevin, 18 (1) (2011) (in press). n [15] C. Mortici, Monotonicity properties of the volume of the ball in R , Optim. Lett. (2010) doi:10.1007/s11590-009-0173-2. [16] Y. Yu, An inequality for ratios of gamma functions, J. Math. Anal. Appl. 352 (2) (2009) 967–970. [17] Y. Yu, A remark on a class of double inequalities of Batir, Expo. Math. 27 (2) (2009) 171–174. [18] W. Burnside, A rapidly convergent series for log N!, Messenger Math. 46 (1917) 157–159. [19] Z. Liu, A new version of the Stirling formula, Tamsui Oxf. J. Math. Sci. 23 (4) (2007) 389–392. [20] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.