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NPS-53BL77021 NAVAL POSTGRADUATE SCHOOL Monterey, California ASYMPTOTIC REPRESENTATION OF STIRLING NUMBERS OF THE SECOND KIND by W. E. Bleick and Peter C. C. Wang // 9 February 1977 I 0A?97 r Approved for public release; distribution unlimited, .86 ] p repared for: ffice of Naval Research (Dr. Bruce McDonald) FEDDOCS tatistics and Probability Branch D 208.1 4/2: NPS-53BL77021 r li n gton, VA 22217 UDLCY KNOX LIBRARY .AVAL POSTGRADUATE SCHOOL :REY, CA 93940 NAVAL POSTGRADUATE SCHOOL Monterey, California Rear Admiral Isham Linder J. R. Borsting Superintendent Provost ABSTRACT: The distribution of the Stirling numbers S(n,k) of the second kind with respect to k has been shown by Harper [Ann. Math. Statist., 38 (1967), 410-414] to be asymptotically normal near the mode. A new single- term asymptotic representation of S(n,k), more effective for large k, is given here. It is based on Hermite's formula for a divided difference and the use of sectional areas normal to the body diagonal of a unit hypercube in k-space. A proof is given that the distribution of these areas is asymptotically normal. A numerical comparison is made with the Harper representation for n=200. This task was supported by: Contracts No. NR-042-286, NSWSES-56953, NISC-56969 NPS-53BL77021 9 February 1977 UNCLASSIFIED SECURITY CLASSl. ICATION OF THIS PAGE (When Data Entered) DUDLEY KNOX U8RARY NAVAL POSTGRADUATE SCHOOt MONTEREY. CA 93940 ^^ REPORT DOCUMENTATION PAGE READ INSTRUCTIONS BEFORE COMPLETING FORM 1. REPORT NUMBER NPS-53BL77021 2. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER 4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED Asymptotic Representation of Stirling Numbers of the Second Kind Technical Report 6. PERFORMING ORG. REPORT NUMBER 7. AUTHORfaj W. E. Bleick Peter C. C. Wang 8. CONTRACT OR GRANT NUMBER^,) NR-042-286 NSWSES-56953 NISC-56969 9. PERFORMING ORGANIZATION NAME AND ADDRESS Naval Postgraduate School Monterey, CA 93940 10. PROGRAM ELEMENT, PROJECT, TASK AREA ft WORK UNIT NUMBERS 11. CONTROLLING OFFICE NAME AND ADDRESS Office of Naval Research (Dr. Bruce McDonald) Statistics and Probability Branch Arlington, VA 22217 12. REPORT DATE 9 February 19 77 13. NUMBER OF PAGES 10 14. MONITORING AGENCY NAME » ADDRESSf// different from Controlling Office) 15. SECURITY CLASS, (of this report) UNCLASSIFIED 15a. DECLASSIFI CATION/ DOWN GRADING SCHEDULE 16. DISTRIBUTION ST ATEMEN T (of this Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, If different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse aide If necessary and Identity by block number) Asymptotic representation Stirling numbers of the second kind Bell number Hermite's formula for a divided difference 20. ABSTRACT (Continue on reverse side It necessary and Identity by block number) The distribution of the Stirling numbers S(n,k) of the second kind with respect to k has been shown by Harper [Ann. Math. Statist., 38 (1967), 410-414] to be asymptotically normal near the mode. A new single-term asymptotic representa- tion of S(n,k), more effective for large k, is given here. It is based on Hermite's formula for a divided difference and the use of sectional areas normal to the body diagonal of a unit hypercube in k-space. A proof is given that the distribution of these areas is asymptotically normal. A numerical | comparison is made with the Harper representation for n=?00. DD FORM 1 JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETE S/N 0102-014- 6601 | UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Data Bntarad) 1. Introduction. Previous asymptotic representations of Stirling numbers S(n,k) of the second kind have been of two types. One type has been a complete infinite series expansion as given by Hsu [1], and by Bleick and Wang [2] and [3]. A second type has been the single-term representation of S(n,k) given by Harper [4] as the normal distribution approximation (1) S(n,k)^— — exp[-(k-y) 2 /2a 2 ] 0-/27 2 where the mean u and the variance a are expressed in terms of the Bell numbers B by n (2) V = B n+1 /B n - 1 and < 3) ° 2 " B n + 2 /B „ - (B n + l /B „ )2 ' X • The purpose of this note is to give a new single-term asymptotic re- presentation based on Hermite's formula for a divided difference, and to compare it with that of Harper. 2. Use of Hermite's formula. A Stirling number S(n,k) of the second kind is defined as the kth difference of z at z=0 divided by kl. By [5, p. 10] we find that this divided difference can be represented by a formula of Hermite as the re- peated definite integral 1 t t . (A) S(n,k) = / dt x / ■ L dt 2 .../ * ± (d K u 1 n /dupdt k where u =t +t^+. .+t . We imagine that t , t„, .., t constitute a set of -1- rectangular Cartesian coordiantes and impose an orthogonal transforma- tion of coordinates to u , u„, .., u, . The volume of the space over which the integration in (4) is performed is a portion of a unit hyper- cube in k-space. If we allow the coordinate u to vary along the body diagonal of the hypercube from at one vertex to k at the opposite vertex, the sectional areas normal to the diagonal cut by the hyper- plane u=t..+t 9 +. •+£, from the domain of integration define a positive function g(u ,k) even with respect to the argument u..-ic/2. We take the integral of g(u..,k) to be k (5) / g(u ,k)du = 1/k! L to agree with the volume of the space over which the integration in (1) is performed. We drop the u.. subscript henceforth. Noting that g(u,k)=0 for k<u<0, we find that (6) g(u,l) = 1 for £ u <_ 1 , (7) 2!g(u,2) = (1 - |u-l|) for £ u £ 2 , and (8) 3!g(i :u,3)={ (3/2-|u-3/2|) 2 /2 for 1/2 < lu-3/2 I : 3/2 3/4 - (u-3/2) 2 for 1 < u < 2 . Consideration of the Laplace transforms of (6), (7) and (8) suggests that we conjecture the Laplace transform of k!g(u,k) to be (9) (l-e- s ) k /s k = e -ks/2 ( sinh s/2 k s/2 for all k. We demonstrate the truth of this conjecture later. On perform- ing the integration in (4) over the variables u , u , .., u we find oo (10) S(n,k) - k!A/ u n_k g(u,k)du . k -2- Using operation 82 of [6, p. 10] on the Laplace transform of u (11) k! / u m g(u,k)du we find the mth moment of the k!g(u,k) distribution about u=0 to be (12) lim (-lAd/ds) m (l-e" S ) k /s k . s-K) It is now easy to demonstrate the truth of the conjecture (9) by show- ing, with the aid of the multinomial theorem, that (12) is the same as the repeated integral 11 1 (13) / dt / dt .../ (t +t +. .+t ) m dt o z R over the volume of the hypercube. Use of (12) and (5) shows the variance of the k!g(u,k) distribution to be (14) a 2 = k/12 . Using (14) the series » t , , 2 2. ns , ks 2 /24 (ks 2 /24) 2 (15) exp (a s /2) = 1 + — =-j + - — ~-\ + • • is the bilateral, but not s multiplied, Laplace transform of the normal distribution (16) (l/a/2T)exp(-t 2 /2a 2 ) according to [7,p.2]]. The corresponding series for (9) multiplied by ks/2 e , or the bilateral Laplace transform of k!g(u,k) shifted left by k/2, is 2 2 2 (17) (2/s) k sinh k S /2 = [1 + SJi + LfUJtl + . . ] k . The dominant k power term in the coefficient of (s /4) in (15) is k /6 n! , and may be shown to be the same in the expansion of (17) by the use of the recurrence formula 6.361 of [8, p. 119]. This proves that the k!g(u,k) distribution is asymptotically normal as k-*». It is remarkable that the normal distribution should arise in the purely -3- geometrical context of sectional areas normal to the body diagonal of a hypercube of high dimension. On replacing k!g(u,k) in (10) by its Gaussian normal approximation 2 of mean u=k/2 and variance a =k/12 we find (18) S(n,k) a, -i — (?) / u n k exp[-(u-k/2) Z /2a Z ]du 2,„ 2 a/2TT ^3k 2,„ 1 / n \ f /i /-> x n-k -t 72. ^ — — (,) J (k/2-at) e dt . /2tT -°° 3. Numerical example. Table 1 compares the exact values of S(200,k) with the asymptotic approximations computed from the single-term representations (1) and o -J (. (18). Harper's representation (1), which uses B =.62475 10 ' u=49.975 and a=3.0551, gives an excellent fit near the mode (k=50) , but (18) gives a much better fit for large values of k. Asymptotic from (1) Table 1. Values of S(200,k) Exact Asymptotic from (18) 2 40 50 60 100 150 199 ,23135 10 39504 10 222 80347 10 69244 10 126 273 24458 10 273 42658 10 273 81579 10 37452 10 275 273 81493 10 .53533 10 275 273 .15285 10 .29658 10 277 274 ,49065 10 217 13938 10 43 .22839 10 30251 10 235 143 .27994 10 .30441 10 235 143 ,16955 10 -241 .19900 10- .19900 10" -4- REFERENCES 1. L. C. Hsu, Note on an asymptotic expansion of the nth difference of zero , Ann. Math. Statist. 19, (1948), 273-277. MR9 , 578. 2. W. E. Bleick and Peter C. C. Wang, Asymptotics of Stirling numbers of the second kind , Proc. Am. Math. Soc. 42 (1974), 575-580. 3. W. E. Bleick and Peter C. C. Wang, Erratum to 2 , Proc. Am. Math. Soc. 48 (1975), 518. 4. L. H. Harper, Stirling behavior is asymptotically normal , Ann. Math. Statist. 38 (1967), 410-414. 5. L. M. Milne-Thomson, The calculus of finite differences , MacMillan and Co., Ltd., London, 1933. 6. G. E. Rober ts and H. Kaufman, Table of Laplace transforms , Saunders, Philadelphia, 1966. MR32 #8050. 7. Balth. van der Pol and H. Bremmer, Operational calculus based on the two-sided Laplace integral , Cambridge University Press, 1955. 8. E. P Adams and R. L. Hippisley, Smithsonian mathematical formulae and tables of elliptic functions , Publication 2672, Smithsonian Institution, Washington, 1922. -5- DISTRIBUTION LIST Copies Copies Statistics and Probability program Office of Naval Research Attn: Dr. B. J. McDonald Arlington, Virgainia 22217 Director, Naval Research Laboratory Attn; Library, Code 2029 (ONRL) Washington, D. 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