Asymptotic Rational Approximation To Pi: Solution of an Unsolved Problem Posed By Herbert Wilf

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1 AofA 0 DMTCS proc AM, 00, Asymptotic Rational Approximation To Pi: Solution of an Unsolved Problem Posed By Herbert Wilf Mark Daniel Ward Department of Statistics, Purdue University, 50 North University St, West Lafayette, IN , USA The webpage of Herbert Wilf describes eight Unsolved Problems Here, we completely resolve the third of these eight problems The task seems innocent: find the first term of the asymptotic behavior of the coefficients of an ordinary generating function, whose coefficients naturally yield rational approximations to π Upon closer examination, however, the analysis is fraught with difficulties For instance, the function is the composition of three functions, but the innermost function has a non-zero constant term, so many standard techniques for analyzing function compositions will completely fail Additionally, the signs of the coefficients are neither all positive, nor alternating in a regular manner The generating function involves both a square root and an arctangent The complex-valued square root and arctangent functions each rely on complex arithms, which are multivalued and fundamentally depend on branch cuts These multiple values and branch cuts make the function extremely tedious to visualize using Maple We provide a complete asymptotic analysis of the coefficients of Wilf s generating function The asymptotic expansion is naturally additive not multiplicative; each term of the expansion contains oscillations, which we precisely characterize The proofs rely on complex analysis, in particular, singularity analysis which, in turn, rely on a Hankel contour and transfer theorems Keywords: analytic combinatorics, asymptotic analysis, rational approximation, generating function, singularity analysis Introduction The webpage of Wilf 009 contains a pdf file of eight Unsolved Problems These include questions about, respectively: series for π; growth of partition functions; a problem in asymptotics; the quadratic character of binomial coefficients; Young tableaux; distinct multiplicities among parts of partitions; Toeplitz determinants; and chromatic number These problems are extremely interesting to read and to think about We completely resolve the third of these eight problems: M D Ward s research was supported by NSF Science & Technoy Center grant CCF c 00 Discrete Mathematics and Theoretical Computer Science DMTCS, Nancy, France

2 59 Mark Daniel Ward A problem in asymptotics Wilf 009 Let fz = arctan e e If fz = n 0 a nz n, find the first term of the asymptotic behavior of the a n s One reason that the behavior of the coefficients of fz is so interesting is that each coefficient can be written in the form a n = b n π c n where b n s and c n s are nonnegative rational numbers For example: n a n 0 4 π 4 π 4 π π π π π π π π π In fact lim n a n /b n = 0 and thus the rational numbers of the form c n /b n provide approximations to π Wilf s rd Unsolved Problem is to find the precise first term asymptotic behavior of the a n s The analysis of fz is surprisingly very challenging, for several reasons: The function fz is the composition of three simpler functions, arctan z z, z, and e Unfortunately, the innermost function, e, has a non-zero constant term In general, when the innermost function in a composition has a non-zero constant term, many of the standard analytic combinatorics techniques completely fail See, for instance, Section VI9, Functional composition, in Flajolet and Sedgewick 009 The coefficients a n are not all positive The signs of the a n s alternate between positive and negative in an erratic, non-systematic way This prevents the analysis of fz using many techniques in enumerative combinatorics that require the coefficients to be all positive or to systematically alternate between positives and negatives The definition of fz involves a square root and an arctangent Each of these functions are defined over the complex numbers using complex arithms, which are multivalued and fundamentally depend on branch cuts These multiple values and choices of branch cuts make the function extremely tedious to visualize using the Maple 4 symbolic algebra software The method of analysis in this paper follows the precept of Painlevé 900 and Hadamard 945: questions posed about real numbers can often be resolved using the methods of complex analysis Flajolet and Sedgewick 009 and Szpankowski 00 give systematic and comprehensive treatments about the use of analytic complex techniques for many types of asymptotic analysis of algorithms, data structures,

3 Asymptotic Rational Approximation To Pi: Solution Of A Wilf Problem 59 and other combinatorial and probabilistic applications We rely on the techniques of singularity analysis, as discussed in Flajolet and Sedgewick 009 Main Results Let gz = e The definition of the square root function in the complex plane is ambiguous, up to a possible change of sign, so we clarify the definition of gz very precisely Let S j = {x iy x, y R with jπ < y < j π} {x jπi x R with x < ln} The regions S j are disjoint and have the following property: L j, where L j is defined as j Z S j = C \ j Z L j = {z = x jπi x R, ln x} In Lemma, we will show that fz has an analytic extension to j Z S j Now we define gz very precisely: { exp gz = e for z S j, where j is an even integer; exp e for z S j, where j is an odd integer; here, we define e unambiguously by the following branch of arithm: Write e = re iθ for r R >0 and 0 θ < π, and then unambiguously define e = ln r iθ For arctan z = i iz iz i, write i = reiθ for r R >0 and π < θ 0, and then define iz i = ln r iθ This choice of the branch of arctan in the numerator of fz allows for an analytic extension to L 0 as well as, automatically, an analytic extension to L j for all even j In contrast, with this choice of branch of arctan, fz cannot be analytically extended to L j for odd j So R = ln πi is the maximum radius around the origin for which fz has an analytic extension, because the closest singularities to the origin will be at ζ = ln πi and its complex conjugate ζ = ln πi The asymptotic analysis of fz using analytic methods relies on having the largest possible disc, centered at the origin, on which fz is analytic So this choice of the branch of arctan is crucial for the analysis that follows later, in Theorem 4 Lemma The function gz = e is analytic on C \ j Z L j Lemma The function fz is analytic on C \ j Z L j Moreover, an appropriate choice for the branch of the arctan function allows fz to have an analytic continuation to j Z L j; however, such an analytic continuation will leave fz discontinuous on j Z L j Figure illustrates the region on which fz is analytic, as in Lemma Theorem Let a n denote the coefficient of z n in the generating function fz = arctan e e, namely, fz = n 0 a nz n Then a n = R n hn, where R = ln πi, and hn is a subexponential factor, ie, lim sup n hn /n =

4 594 Mark Daniel Ward Y 0π 8π 6π 4π π Y Axis X -π -4π -6π -8π -0π Fig : Region of the complex plane for which fx iy is analytic The closest singularities to the origin are located at ζ = ln πi and its complex conjugate ζ = ln πi So the radius of convergence is R = ζ, and C R := { z z < R } is the largest open disc around the origin on which fz is analytic X Axis Rigorous proofs of Lemma, Lemma, and Theorem are given in Section The veracity of Theorem is also evident in Figure Using a n as in Theorem, the Maple plot in Figure reinforces our understanding that lim sup n hn /n = To fully complete the solution of Wilf s problem, we Fig : Maple plot of hn /n = R a n /n = ln πi [z n ]fz /n for 00 n 000 must precisely describe the first term of the asymptotic behavior of hn too As we see in Theorem 4, a n is oscillating; this is also apparent in the figures Theorem 4 gives the complete description of the asymptotic behavior of the a n s We follow the notation from pages 8 84 of Flajolet and Sedgewick 009 in the definitions of e k and λ k,l Theorem 4 Let a n denote the coefficient of z n in the generating function fz = arctan e e, namely, fz = n 0 a nz n Then for each positive integer N, N a n = π j=0 N j cosn j / arctanπ/ ln d j Γ j /R n j/ n j/ k= e k j n k O R n n N/ where d j = [z j / ]e z /, and e k α = k l=k l λ k,l α α α l, and λ k,l = [v k t l ]e t vt /v, and R = ln πi,

5 595 Asymptotic Rational Approximation To Pi: Solution Of A Wilf Problem For instance, when N = in Theorem 4, cosn / arctanπ/ ln / an = π n 8n Rn/ 8n5/ cosn / arctanπ/ ln Rn / 8n/ 64n5/ cosn / arctanπ/ ln Rn / 84n5/ O Rn n7/ We prove Theorem 4 in Section In addition to a formal proof, we include several figures that give credence to Theorem 4 Fig : Maple plot of n Rn/ an π for 00 n 000 π Figure depicts the fact that the first additive term of the asymptotic behavior of an is n Rn/ multiplied by some oscillating term, ie, Theorem 4 gives the correct additive first order of an For other examples of such oscillations in asymptotic terms, see for instance: the section Nonperiodic fluctuations pages or note VI4 pages both from Flajolet and Sedgewick 009 Figure 4 exhibits the normalized difference between an and the first three terms of the asymptotic expansion, from Theorem 4 Figures 5 and 6 illustrate the fact that the asymptotic expansion of an must be expressed a additively, not multiplicatively In other words, it is not true that cosn/narctanπ/ ln π n Rn/ Since the cosine is occasionally extremely close to 0 as seen by the spikes in Figures 5 and 6, the lower order terms of the asymptotic expression will occasionally dominate the otherwise higher order terms These figures help provide insight into the asymptotic properties of an Proofs Proof: of Lemma For z in the interior of Sj, the function gz = e is the composition of analytic functions and is thus also analytic So we only need to check that f z is continuous and analytic on the boundary between Sj and Sj, ie, for z = x jπi with x R<ln We claim lim gt = gz t z t Sj and lim g 0 t = g 0 z t z t Sj

6 596 Mark Daniel Ward cosn/ arctanπ/ ln Fig 4: Maple plot of R n π n/ / 5/ n R 8n 8n cosn / arctanπ/ ln ln 64n5/ cosn /Rarctanπ/ an for 00 n 000 n / Rn / 8n/ 84n5/ n 7/ Fig 5: Maple plot of an cosn/ arctanπ/ ln π n/ for 00 n 000 nr ln Maple plot of an divided by π cosn/rarctanπ/ n/ / 5/ n 8n 8n cosn / arctanπ/ ln ln 64n5/ cosn /Rarctanπ/ for 00 n 000 n / Rn / 8n/ 84n5/ Fig 6:

7 Asymptotic Rational Approximation To Pi: Solution Of A Wilf Problem 597 These properties are already clear for t S j, so we only check for t S j Define r, r t R >0 and θ R 0 such that e t = r t e iθt and e = r R >0 Then r t r and θ t 0, so e t = lnr t iθ t lnr 0i Thus exp e t exp lnr = r So gt = e t = exp e t r = gz, for j even; gt = e t = exp e t r = gz, for j odd Thus gz is continuous at z Since g z = e gz /e, then g t g z Thus, gz is indeed analytic on C \ j Z L j Proof: of Lemma The set {z = iy y R, y } is sent by the conformal map z iz i to the line {z = x x R 0 }; otherwise the image of z iz i is disjoint from {z = x x R 0 } Therefore, iz i is analytic on the set z / {z = iy y R, y } ie, the branch cut is avoided Since gz is analytic for z C \ j Z L j shown in Lemma, and the range of gz is arctan gz gz igz i gz gz disjoint from {z = iy y R, y }, it follows that fz = = i is the composition of analytic functions and is thus also analytic for z C \ j Z L j Let j be an even integer We prove that fz has an analytic continuation to L j Each z L j can be written in the form z = x jπi for x R ln Then e R 0 We define fz = ln e e e for z = x jπi where x R >ln ; and also fln jπi = Consider z S j with z ln jπi If t S j and with t z, then gt i e so ft = i igt i gt gt ln e e e = ln e e e Similarly, for t S j with t z, we have gt i e so ft = i igt i gt gt ln e e e Also, fz is continuous at z = ln jπi, and the derivative of fz exists and is continuous all of this is straightforward to check, so the definition in is an analytic continuation of fz for z L j For odd j, an additional analytic continuation of fz to L j is not possible Consider z = x jπi for x R with x > ln Then e R <0 Thus, for t S j with t z, we have gt i e so ft = i igt i gt gt ln e π e = e ln e π e e

8 598 Mark Daniel Ward Similarly, for t S j with t z, we have gt i e so ft = i igt i gt gt ln e π e e = ln e π e e Thus, fz is not continuous, and thus not analytic, at z = x jπi for odd integers j and x > ln Similarly, fz is discontinuous at ln jπi for odd j Proof: of Theorem By Lemma, the radius of convergence of fz about the origin is exactly R = ζ = ln πi Indeed, on the boundary of C R, the function fz has non-removable singularities exactly at z = ζ = ln πi and z = ζ = ln πi So the radius of convergence of fz about the origin is exactly R The classical Exponential Growth Formula see, eg, Flajolet and Sedgewick 009 provides a natural correspondence between the radius of converge of any function fz and the rate of growth of the function s coefficients: If fz is analytic at all points in an open disc of radius r about the origin, but has a non-removable singularity on the boundary, then the coefficients a n = [z n ]fz satisfy a n = r n hn, where h is a subexponential factor, ie, lim sup n hn /n = Proof: of Theorem 4 We use the theory of singularity analysis from Flajolet and Sedgewick 009 In particular, we use Theorem VI5 from page 98 of Flajolet and Sedgewick 009, which we quote: Theorem VI5 Singularity analysis, multiple singularities Flajolet and Sedgewick 009 Let fz be analytic in z < ρ and have a finite number of singularities on the circle z = ρ at points ζ j = ρe iθj, for j = r Assume that there exists a -domain 0 such that fz is analytic in the indented disc r D = ζ j 0, j= with ζ 0 the image of 0 by the mapping z ζz Assume that there exists r functions σ,, σ r, each a linear combination of elements from the scale S, and a function τ S such that fz = σ j z/ζ j Oτz/ζ j as z ζ j in D Then the coefficients of fz satisfy the asymptotic estimate f n = r j= ζ n j σ j,n Oρ n τn, where each σ j,n = [z n ]σ j z has its coefficients determined by Theorems VI, VI and τ n = n a n b, if τz = z a λz b For the function fz = arctan e e in Wilf s problem, fz is analytic in the open disc C R := { } z z < R On the boundary of CR, the function fz has non-removable singularities at z = ζ, ζ In

9 Asymptotic Rational Approximation To Pi: Solution Of A Wilf Problem 599 the notation in Flajolet and Sedgewick 009, we have ζ = ζ and ζ = ζ, as well as ρ = R We define the -domain { } 0 = z ln 6πi z <, z, argz > arctanπ/ ln ln πi By using the domain in which fz is analytic, established in Lemma, we see that fz is analytic throughout the domain D = ζ 0 ζ 0 Fix a positive integer N Then we can define π σ m z/ζ m = N j= ζ m z j /j! τz = z N /, and we have the required conditions for m =, ie, for ζ = ζ and ζ = ζ, fz = σ m z/ζ m Oτz/ζ m as z ζ m in D, for m =, Let d j = [z j / ]e z /, so {d 0, d, d, } = {, 4, 96, } Note that ez / is the square root of an exponential generation function for the Bernoulli numbers Then N σ m z/ζ m = π d j ζ m z j / Oζ m z N /, for m =, j=0 For instance, if N =, σ m z/ζ m = π ζ m z / 4 ζ m z / 96 ζ m z / Oζ m z 5/, for m =, The explicit asymptotic expansions z α are also well-known we follow the notation from pages 8 84 of Flajolet and Sedgewick 009, [z n ] z α nα e k α Γα n k, where e k α is a sum of polynomials in α of degrees between k and k thus, the overall degree of e k α is k, e k α = k l=k k= l λ k,l α α α l, and λ k,l = [v k t l ]e t vt /v Thus, the coefficients a n of fz satisfy the asymptotic estimate N ζ nj / ζ nj / a n = π d j Γ j /n j/ j=0 N j k= e k n k O R n n N/

10 600 Mark Daniel Ward In the case N =, we explicitly compute the asymptotic expansions: [z n ] z / = πn 8n 8n On ; [z n ] z / = πn 6n On ; [z n ] z / = πn 5 4 On It follows that the coefficients a n of fz satisfy the asymptotic estimate a n = [ π ζ n n/ ζ n/ 8n / 8n 5/ ζ n / ζ n / 8n ] / 64n 5/ ζ n / ζ n / 84n 5/ O R n n 7/ To simplify these expressions, we note that, for a = n j /, So it follows that R a ζ a ζ a = ζ/ζ a/ ζ/ζ a/ a = exp ζ/ζ exp a ζ/ζ = exp ia i π i ln i π exp ia i π i ln ln i π ln = exp ia arctanπ/ ln expia arctanπ/ ln = cosa arctanπ/ ln N cosn j / arctanπ/ ln a n = π d j Γ j /R n j/ n j/ j=0 N j k= e k j n k O For example, when N =, a n = [ cosn / arctanπ/ ln π n R n/ 8n / 8n 5/ cosn / arctanπ/ ln R n / 8n / 64n ] 5/ cosn / arctanπ/ ln R n / 84n 5/ O R n n 7/ R n n N/ For more terms in the asymptotic expansion, the method of analysis proceeds in exactly the same way

11 Asymptotic Rational Approximation To Pi: Solution Of A Wilf Problem 60 Acknowledgements The author sincerely thanks Herb Wilf for posting this very interesting problem on his webpage, for a short conversation about the problem at AofA 009, and for insightful discussions during the author s visit to the University of Pennsylvania after submitting the solution The author is very thankful for discussions prior to submission with a few colleagues about this problem, namely, Steve Bell, Michael Drmota, Philippe Flajolet, Bruno Salvy, and Wojciech Szpankowski The author also warmly thanks Pierre Nicodème and Mireille Régnier, who graciously hosted the author during his recent visit to Paris in November 009 Between the initial submission and the final version for publication, the author was fortunate to get several insightful questions and suggestions from Alex Eremenko, Andrei Gabrielov, and Robin Pemantle The author also appreciates constructive, helpful advice garnered from three anonymous reviewers The author is thankful to all of these people for the resulting improvements to the content and presentation of the paper References L V Ahlfors Complex Analysis McGraw-Hill, New York, rd edition, 979 J W Brown and R V Churchill Complex Variables and Applications McGraw-Hill, 8th edition, 009 P Flajolet and R Sedgewick Analytic Combinatorics Cambridge, 009 J Hadamard The Psychoy of Invention in the Mathematical Field Princeton, 945 P Henrici Applied and Computational Complex Analysis volumes Wiley, New York, 974, 977, 986 P Painlevé Analyse des travaux scientifiques jusqu en 900 Gauthier-Villars, 900 W Szpankowski Average Case Analysis of Algorithms on Sequences Wiley, New York, 00 H Wilf Some unsolved problems, September 009 Available for download as an electronic pdf file, wilf/website/unsolvedproblemspdf

12 60 Mark Daniel Ward