Variations on a Theme by James Stirling

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1 Variations on a Theme by James Stirling Diego Dominici Department of Mathematics State University of New York at New Paltz 75 S. Manheim Blvd. Suite 9 New Paltz, NY USA Phone: (845) Fax: (845) Dedicated to Donald Silberger on the occasion of his 76th birthday 1 Introduction The most widely known and used result in asymptotics is probably Stirling s formula, n! πnn n e n, n (1) named after James Stirling (May Dec. 1770). The formula provides an extremely accurate approximation of the factorial numbers for large values of n. The asymptotic formula (1), for which Stirling is best known, appeared as Example to Proposition 8 in his most important work, Methodus Differentialis, published in 1730 [41]. In it, he asserted that log (n!) is dominicd@newpaltz.edu 1

2 approximated by three or four terms of the series ( n + 1 ) ( log n + 1 ) ( a n + 1 ) + 1 log (π) () a 4 ( 7a ) + n ( ) n + 1 3, where log means the base-10 logarithm and a = [ln(10)] 1 [40]. In 1730 Stirling wrote to Abraham De Moivre (6 May Nov. 1754) pointing out some errors that he had made in a table of logarithms of factorials in the book and also telling him about (). After seeing Stirling s results, De Moivre derived the formula ( ln [(n 1)!] n 1 ) ln(n) n + 1 ln (π)+ B k, (3) k (k 1) nk 1 k 1 which he published in his Miscellaneis Analyticis Supplementum a few months later. Equation (3) is called Stirling s series and the numbers B k are called the Bernoulli numbers, and are defined by k ( ) k +1 B 0 =1, B j =0, k 1. (4) j j=0 Clearly Stirling and De Moivre regularly corresponded around this time, for in September 1730 Stirling relates the new results of De Moivre in a letter to Gabriel Cramer. In 179 Leonhard Euler (15 April Sept. 1783) proposed a generalization of the factorial function from natural numbers to positive real numbers [11]. It is called the gamma function, Γ(z), which he defined as n!n z Γ(z) = lim n z(z +1) (z + n), (5) and it is related to the factorial numbers by Γ(n +1)=n!, n =0, 1,,... From Euler s definition (5), we immediately obtain the fundamental relation Γ(z +1)=zΓ(z) (6) and the value Γ(1) = 1. In fact, the gamma function is completely characterized by the Bohr-Mollerup theorem [8]:

3 Theorem 1 The gamma function is the only function Γ:(0, ) (0, ) which satisfies 1. Γ(1) = 1. Γ(x +1)=xΓ(x) 3. ln [Γ(x)] is convex for all x (0, ). Proof. See [4] and [30]. Another complex-analytic characterization is due to Wielandt [43]: Theorem The gamma function is the only holomorphic function in the right half plane A satisfying 1. Γ(1) = 1. Γ(z +1)=zΓ(z) for all z A 3. Γ(z) is bounded in the strip 1 Re(z) < Proof. See [35]. In terms of Γ(z), we can re-write (1) as ln [Γ(z)] P(z), z (7) with P(z) =z ln (z) z 1 ln (z)+1 ln (π) (8) and (3) in the form where R 0 (z) = 0 and ln [Γ(z)] P(z)+R N (z), z (9) R N (z) = N k=1 B k, N 1. (10) k (k 1) zk 1 Estimations of the remainder ln [Γ(z)] P(z) R N (z) were computed in [38]. 3

4 Previous results Over the years, there have been many different approaches to the derivation of (7) and (9), including: 1. Aissen [] studied the sequence V n = nn e n n!. Using his lemma Lemma 3 If y n+1 =1+ α y n n + O ( n ) and y n 0for all n, then y n Cn α, n for some non-zero constant C. he showed that n! C nn n e n.. Bender & Orszag [5], Bleistein & Handelsman [6], Diaconis & Freedman [13], Dingle [14], Olver [33] and Wong [45] applied Laplace s method to the Euler integral of the second kind Γ(z) = 0 t z 1 e t dt, Re(z) > 0. (11) 3. Bender & Orszag [5] and Temme [39] used Hankel s contour integral [1] 1 Γ(z) = 1 πi and the method of steepest descent. (0+) t z e t dt 4. Bleistein & Handelsman [6], Lebedev [5], Sasvári [37] and Temme [39] used Binet s first formula and ln [Γ(z)] = P(z) t ( 1 1 t 1 t + 1 ) e t 1 ( 1 1 t + 1 ) e tz dt, Re(z) > 0 e t 1 B k = (k)! tk, t < π. k 1 4

5 5. Blyth & Pathak [7] and Khan [3] used probabilistic arguments, applying the Central Limit Theorem and the limit theorem for moment generating functions to Gamma and Poison random variables. 6. Coleman [9] defined c n = ( n + 1 ) ln(n) n +1 ln (n!), and showed that c n 1 1 ln (π) asn. A similar result was proved by Aissen [], using the concavity of ln(x). 7. Dingle [14] used Weierstrass infinite product 1 Γ(z) = zeγz n=1 [( 1+ z ) ] e z n, n (where γ is Euler s constant) and Mellin transforms. 8. Feller [17], [18] proved the identity where ln (n!) 1 ( ) 1 ln(n) =I(n) I n 1 + (a k b k )+a n, (1) k=1 n k I(n) = ln(t)dt, a k = 0 k 1 ln ( ) k dt, b k = t k+ 1 k ln ( ) t dt k and showed that (a k b k ) I k=1 ( ) 1 = 1 ln (π). There is a mistake in his Equation (.4), where he states that instead of (1). ln (n!) 1 ( ) 1 ln(n)+i(n) I n 1 = (a k b k )+a n, 5 k=1

6 9. Hayman [19] used the exponential generating function e z 1 = k! zk k=0 and his method for admissible functions. 10. Hummel [0], established the inequalities where 11 1 <r n + 1 ln (π) < 1, n =, 3,..., ( ) n!e n r n =ln. πnn n Impens [1], [], showed that for x>0 R n (x) < ln [Γ(x)] P(x) <R m+1 (x), n,m 0, where R n (x) was defined in (10). Maria [6] showed that [ ] 1 3 1n + <r n, n =1,,... (n +1) Mermin [8] proved the identity ( e rn = e ) k+ 1, k k=n which he used to show that r n R 3 (n). Michel [9], proved the inequality 1 1 ern 1n 1 88n 1 360n + 1, n =3, n4 Nanjundiah [3], showed that R (n) <r n <R 1 (n) =1,,... Robbins [36], established the double inequality 1 1n +1 <r n < 1, n =1,,... 1n 6

7 11. Marsaglia & Marsaglia [7] derived from (11) the asymptotic expansion ( ) k ( ) n! n n+1 e n k b k Γ k, n k=1 where the generating function G(z) = k 0b k z k is defined by G(z) exp [1 G(z)] = exp ( 1 ) z, G (0) = Namias [31] introduced the function F (n) = Γ(n), with P(n) defined in P(n) (8). From Legendre s duplication formula Γ(n) = n 1 π Γ(n)Γ ( n + 1 ), (13) he derived a functional equation for F (n) F (n) F (n)f ( ) = ( e 1 1 ) n, n 1 n from which he obtained (9). He also considered the triplication case, using Gauss multiplication formula ) m 1 1 m ( Γ(mz) =( π m mz 1 Γ z + k ), m =, 3,... m k=0 with m =3. His results where extended by Deeba & Rodriguez in [1]. 13. Olver [33] used Euler s definition (5) ln [Γ(z)] = ln (z) + lim n z ln (n)+ and the Euler-Maclaurin formula. Knopp [4] and Wilf [44]. n [ln (k) ln (z + k)] k=1 A similar analysis was done by 14. Patin [34] used (11) and the Lebesgue Dominated Convergence Theorem. 7

8 15. Whittaker & Watson [4] used Binet s second formula and ln [Γ(z)] = P(z) + 0 arctan ( ) t z dt, Re(z) > 0 e πt 1 ( 1) k arctan(x) =x k +1 xk, x 1. k=0 Thus, there have been a huge variety of approaches to Stirling s result, ranging from elementary to heavy-machinery methods. In an effort to join such illustrious company, we present still another direction for deriving (9). Our starting point shall be the difference equation (6). A parallel approach was considered in [3]. For a different analysis of (6) using the method of controlling factors, see [5]. Extensions and other applications of the method used can be found in [10], [15] and [16]. 3 Asymptotic analysis 3.1 Stirling s formula We begin with a derivation of (7), to better illustrate how the method works. We assume that ln [Γ(z)] f(z)+g(z), z (14) with g = o(f), z. (15) Using (14) in (6), we have f(z +1) f(z)+g(z +1) g(z) ln(z). (16) Expanding f(z + 1) and g(z + 1) in a Taylor series, we obtain From (15) and (17) we get the system f (z)+ 1 f (z)+g (z) ln(z). (17) f (z) = ln(z), 1 f (z)+g (z) =0 8

9 and thus, f(z) =z ln(z) z, g(z) = 1 ln(z)+c. (18) To find the constant C in (18), we replace Γ(z) e f(z)+g(z) in (13) and obtain e f(z)+g(z) z 1 π e f(z)+g(z) e f(z+ 1 )+g(z+ 1 ), or e C 1 ( 1+ 1 z ) z π, z, from which we conclude that C =ln ( π ). Hence, we have shown that ln [Γ(z)] z ln(z) z 1 ln(z)+1 ln (π), z. 3. Stirling s series To extend the result of the previous section, we now assume that N ln [Γ(z)] f k (z), z (19) k=0 with f k+1 = o (f k ), z, k =0, 1,...,N 1. (0) Using (19) in (6) we have N f k (z +1) f k (z) ln (z), z. (1) k=0 Replacing the Taylor series f k (z +1)= 1 d j j! dz f k(z) j j 0 in (1), we have N 1 d j j! dz f k(z) ln (z), z. () j k=0 j 1 9

10 and From (0), we obtain the system of ODEs which imply and k 1 j=0 d dz f 0 =ln(z) d k+1 j 1 (k +1 j)! dz f j(z)+ d k+1 j dz f k =0, k 1, f k (z) = f 0 (z) =z ln(z) z (3) k 1 d j (j + 1)! dz f k j(z), k 1, (4) j j=1 where we have omitted (for the time being) any constant of integration. To find the functions f k (z), we set in (4) and get a 0 = 1 and d k f k (z) =a k dz f 0(z) (5) k which gives or a k d k dz k f 0(z) = a 0 =1, a 0 =1, k 1 d j (j +1)! dz j j=1 a k = d k j dz k j f 0(z), k 1, k 1 (j +1)! a k j, k 1, j=1 k 1 (k +1 j)! a j =0, k 1. (6) j=0 Multiplying both sides by (k + 1)!, we can write (6) as a 0 =1, k (k + 1)! (k +1 j)!j! j!a j =0, k 1, j=0 10

11 or a 0 =1, k ( ) k +1 j!a j =0, k 1. (7) j j=0 Comparing (4) and (7) we conclude that a k = B k, k! k 0. (8) Thus, from (3), (5) and (8) we have from which we obtain and f k (z) = B k k! d k [z ln(z) z], k 0, dzk f 1 (z) = 1 ln(z) (9) f k (z) = B k (k 1)! k! ( 1)k = ( 1)k B k, k. z k 1 k (k +1)zk 1 Since B k+1 = 0 for all k 1, we need to consider even values of k only, B k f k (z) = k (k +1)zk 1, k 1. (30) So far, we haven t included any constant of integration in our calculations. We could add a constant to one of the functions f k (z), let s say to f 1 (z), and proceed as in Section 3.1 to find it. Doing this, we would obtain from (3), (9) and (30) that N f k (z) =P(z)+R N (z) k=0 where P(z), R N (z) were defined in (8) and (10) respectively. Another possibility, would be to assume no previous knowledge of Γ(z), except for the difference equation Γ(z +1) = zγ(z) and the value at 1, Γ(1) = 1. In doing so, (19) would imply the initial conditions f k (1) = 0, for all k 0. Hence, we would have f 0 (z) =z ln(z) z +1, f 1 (z) = 1 ln(z) f k (z) = B k k (k +1)z B k k 1 k (k +1), k 1 11

12 and therefore ln [Γ(z)] z ln(z) z 1 ln(z)+c N + R N (z), (31) with N B k C N =1 k (k +1). (3) k=1 Computing the first few C N, we would get C , C.91944, C , C , C , C 6.903, C , C , C , C , and increasingly greater numbers (in absolute value). We would conclude that, before the sum starts diverging, the C N seem to be approaching a value close to.918. Given our previous discussion of Stirling s formula, it is not surprising to find that 1 ln (π) Of course, geniuses like Euler or Gauss would reach the conclusion that the optimal constant equals 1 ln (π) without knowing anything about Stirling s work! 4 Conclusion We have presented the history and previous approaches to the proof of Stirling s series (9). We have used a different procedure, based on the asymptotic analysis of the difference equation (6). The method reproduces (9) very easily and can be extended to use in more complicated difference equations. Bender and Orszag observed in [5, Page 7] that without further information the constant 1 ln (π) cannot be determined. The difference equation that we have solved is linear and homogeneous, so any arbitrary multiple of a solution is still a solution. While agreeing with them completely, we have shown that by imposing the additional condition Γ(1) = 1 one can find an approximation to the value of 1 ln (π), without any other assumptions. Thus, local behavior at z =1 and asymptotic behavior as z can be combined fruitfully. We sincerely hope that more and more professors and their students will discover the beauty contained in the (very!) Special Functions, among which Γ(z) is, without doubt, a prima donna. 1

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