The distribution of the Stirling numbers S(n,k) of the second kind with respect to k has been shown 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=200Office of Naval Research (Dr. Bruce McDonald), Statistics and Probability Branch, Arlington, VAhttp://archive.org/details/asymptoticrepres00bleiNR-042-286, NSWSES-56953, NISC-56969N

Bleick, Willard Evan

Wang, Peter C.C.

The distribution of the Stirling numbers S(n,k) of the second kind with respect to k has been shown 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=200NR-042-286, NSWSES-56953, NISC-56969NAOffice of Naval Research (Dr. Bruce McDonald), Statistics and Probability Branch, Arlington, VAhttp://archive.org/details/asymptoticrepres00ble

Calhoun, Institutional Archive of the Naval Postgraduate School

NPS-53BL77021NAVAL POSTGRADUATE SCHOOLMonterey, CaliforniaASYMPTOTIC REPRESENTATION OFSTIRLING NUMBERS OF THE SECOND KINDbyW. E. Bleick and Peter C. C. Wang//9 February 1977I 0A?97 r Approved for public release; distribution unlimited,.86] prepared for:ffice of Naval Research (Dr. Bruce McDonald)FEDDOCS tatistics and Probability BranchD 208.1 4/2: NPS-53BL77021 rlington, VA 22217UDLCY KNOX LIBRARY.AVAL POSTGRADUATE SCHOOL:REY, CA 93940NAVAL POSTGRADUATE SCHOOLMonterey, CaliforniaRear Admiral Isham Linder J. R. BorstingSuperintendent ProvostABSTRACT:The distribution of the Stirling numbers S(n,k) of the second kindwith 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, isgiven here. It is based on Hermite's formula for a divided differenceand the use of sectional areas normal to the body diagonal of a unithypercube in k-space. A proof is given that the distribution of theseareas is asymptotically normal. A numerical comparison is made with theHarper representation for n=200.This task was supported by: Contracts No. NR-042-286,NSWSES-56953,NISC-56969NPS-53BL770219 February 1977UNCLASSIFIEDSECURITY CLASSl. ICATION OF THIS PAGE (When Data Entered)DUDLEY KNOX U8RARYNAVAL POSTGRADUATE SCHOOtMONTEREY. CA 93940^^REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORM1. REPORT NUMBERNPS-53BL770212. GOVT ACCESSION NO 3. RECIPIENT'S CATALOG NUMBER4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVEREDAsymptotic Representation ofStirling Numbers of the Second KindTechnical Report6. PERFORMING ORG. REPORT NUMBER7. AUTHORfajW. E. BleickPeter C. C. Wang8. CONTRACT OR GRANT NUMBER^,)NR-042-286NSWSES-56953NISC-569699. PERFORMING ORGANIZATION NAME AND ADDRESSNaval Postgraduate SchoolMonterey, CA 9394010. PROGRAM ELEMENT, PROJECT, TASKAREA ft WORK UNIT NUMBERS11. CONTROLLING OFFICE NAME AND ADDRESSOffice of Naval Research (Dr. Bruce McDonald)Statistics and Probability BranchArlington, VA 2221712. REPORT DATE9 February 19 7713. NUMBER OF PAGES1014. MONITORING AGENCY NAME » ADDRESSf// different from Controlling Office) 15. SECURITY CLASS, (of this report)UNCLASSIFIED15a. DECLASSIFI CATION/ DOWN GRADINGSCHEDULE16. 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 NOTES19. KEY WORDS (Continue on reverse aide If necessary and Identity by block number)Asymptotic representationStirling numbers of the second kindBell numberHermite's formula for a divided difference20. 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 respectto k has been shown by Harper [Ann. Math. Statist., 38 (1967), 410-414] to beasymptotically 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 onHermite's formula for a divided difference and the use of sectional areasnormal to the body diagonal of a unit hypercube in k-space. A proof is giventhat the distribution of these areas is asymptotically normal. A numerical| comparison is made with the Harper representation for n=?00.DD FORM1 JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETES/N 0102-014- 6601 |UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGE (When Data Bntarad)1. Introduction.Previous asymptotic representations of Stirling numbers S(n,k) ofthe second kind have been of two types. One type has been a completeinfinite series expansion as given by Hsu [1], and by Bleick and Wang[2] and [3]. A second type has been the single-term representation ofS(n,k) given by Harper [4] as the normal distribution approximation(1) S(n,k)^—— exp[-(k-y) 2 /2a 2 ]0-/272where the mean u and the variance a are expressed in terms of the Bellnumbers B byn(2) V = Bn+1/Bn- 1and< 3) °2"Bn+2/B„-(Bn+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 tocompare 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 kthdifference of z at z=0 divided by kl. By [5, p. 10] we find that thisdivided difference can be represented by a formula of Hermite as the re-peated definite integral1 t t .(A) S(n,k) = / dtx/Ldt2.../ *±(dKu1n/dupdtkwhere 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 overwhich 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 bodydiagonal of the hypercube from at one vertex to k at the oppositevertex, the sectional areas normal to the diagonal cut by the hyper-plane u=t..+t 9+. •+£, from the domain of integration define a positivefunction g(u ,k) even with respect to the argument u..-ic/2. We take theintegral of g(u..,k) to bek(5) / g(u ,k)du = 1/k!Lto 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)=0for 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 : 3/23/4 - (u-3/2) 2 for 1 < u < 2 .Consideration of the Laplace transforms of (6), (7) and (8) suggests thatwe conjecture the Laplace transform of k!g(u,k) to be(9) (l-e- s )k/sk= e-ks/2(sinh s/2 ks/2for all k. We demonstrate the truth of this conjecture later. On perform-ing the integration in (4) over the variables u , u , .., u we findoo(10) S(n,k) - k!A/ un_kg(u,k)du.k-2-Using operation 82 of [6, p. 10] on the Laplace transform ofu(11) k! / umg(u,k)duwe 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 /sk .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 asthe repeated integral 11 1(13) / dt / dt .../ (t +t +. .+t )mdtoz Rover the volume of the hypercube.Use of (12) and (5) shows the variance of the k!g(u,k) distributionto be(14) a2= k/12 .Using (14) the series» t , , 2 2. ns , ks2/24 (ks 2 /24)2(15) exp (a s /2) = 1 + —=-j + -—~-\ + • •is the bilateral, but not s multiplied, Laplace transform of thenormal distribution(16) (l/a/2T)exp(-t 2 /2a 2 )according to [7,p.2]]. The corresponding series for (9) multiplied byks/2e , or the bilateral Laplace transform of k!g(u,k) shifted left byk/2, is2 2 2(17) (2/s)ksinhkS /2 = [1 + SJi + LfUJtl + . . ] k .The dominant k power term in the coefficient of (s /4) in (15) isk /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 provesthat the k!g(u,k) distribution is asymptotically normal as k-*». It isremarkable that the normal distribution should arise in the purely-3-geometrical context of sectional areas normal to the body diagonal ofa hypercube of high dimension.On replacing k!g(u,k) in (10) by its Gaussian normal approximation2of mean u=k/2 and variance a =k/12 we find(18) S(n,k) a, -i—(?) / un kexp[-(u-k/2) Z /2a Z ]du2,„ 2a/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 asymptoticapproximations computed from the single-term representations (1) ando-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.Asymptoticfrom (1)Table 1. Values of S(200,k)ExactAsymptoticfrom (18)2405060100150199,23135 1039504 10222 80347 10 69244 10126273 24458 10273 42658 1027381579 1037452 1027527381493 10.53533 10275273.15285 10.29658 10277274,49065 1021713938 1043.22839 1030251 10235143.27994 10.30441 10235143,16955 10-241.19900 10- .19900 10"-4-REFERENCES1. L. C. Hsu, Note on an asymptotic expansion of the nth differenceof zero,Ann. Math. Statist. 19, (1948), 273-277. MR9 , 578.2. W. E. Bleick and Peter C. C. Wang, Asymptotics of Stirling numbersof 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 , MacMillanand 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 onthe two-sided Laplace integral , Cambridge University Press, 1955.8. E. P Adams and R. L. Hippisley, Smithsonian mathematical formulaeand tables of elliptic functions , Publication 2672, SmithsonianInstitution, Washington, 1922.-5-DISTRIBUTION LISTCopies CopiesStatistics and ProbabilityprogramOffice of Naval ResearchAttn: Dr. B. J. McDonaldArlington, Virgainia 22217Director, Naval ResearchLaboratoryAttn; Library, Code 2029(ONRL)Washington, D. 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Asymptotic Representation of Stirling Numbers of the Second Kind

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Asymptotic Representation of Stirling Numbers of the Second Kind

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