Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.bg.ac.yu Appl. Anal. Discrete Math. x (xxxx), xxx xxx. doi:10.2298/aadmxxxxxxxx ON THE COEFFICIENTS OF AN ASYMPTOTIC EXPANSION RELATED TO SOMOS QUADRATIC RECURRENCE CONSTANT Gergő Nemes In this paper we study the coefficients of an asymptotic expansion related to Somos Quadratic Recurrence Constant. We develop recurrence relations and an asymptotic estimation for them. We also present some results which show that these coefficients are related to the Ordered Bell Numbers. 1. INTRODUCTION Somos Quadratic Recurrence Constant, named after M. Somos [2, p. 446] [5], is the number σ = 1 2 3 4 = 1.6616879496.... The constant σ arose when he had examined the asymptotic behavior of the sequence (1) g 0 = 1, g n = ng 2 n 1, n 1, with first few terms 1, 1, 2, 12, 576, 1658880,.... asymptotic series as follows: (2) σ 2n ng n a 0 + a 1 n + a 2 n 2 + a 3 n 3 + a 4 n 4 +, n +, Somos showed that g n has an where the first few a coefficients are 1, 2, 1, 4, 21, 138,... (this is sequence A116603 in the On-line Encyclopedia of Integer Sequences). From (1) it can be shown that the generating function A (x) = 0 a x 2000 Mathematics Subject Classification: 05A15; 05A16; 11B37 Keywords and Phrases: asymptotic approximation; generating functions; recurrence relations 1
2 G. Nemes satisfies the functional equation (3) A 2 (x) = () 2 A x. J. Sondow and P. Hadjicostas [6] showed that for x 0 (4) A (x) = 1 () 1/2. The purpose of this paper is to develop recurrence relations and an asymptotic approximation for the a s. 2. MAIN RESULTS Our first theorem gives a recurrence relation for the a coefficients in the asymptotic expansion (2). for 3. Theorem 1. The a coefficients satisfy the recurrence relation 1 ( a 0 = 1, a 1 = 2, a 2 = 1, a = ( 1) j 3 ) a j, In the second theorem we show that the generating function A (x) can be expressed in terms of the Ordered Bell Numbers [7, p. 189], i.e., the number of ordered partitions of the set {1,..., n}. The Ordered Bell Numbers b have the following exponential generating function 1 2 e x = 0 They are given explicitly by the formula From (4) we will prove b = j 0 b! x. j 2 j+1. Theorem 2. The generating function of the a coefficients has the following representation (5) A (x) = exp ( 1) 1 2b x. 1 Note that this is only a formal expression due to the rapid growth of the b s (see formula (10)). As a corollary we represent the generating function of the Ordered Bell Numbers in terms of the logarithmic derivative of A (x).
On the coefficients of an asymptotic expansion 3 Corollary 1. The generating function of the Ordered Bell Numbers has the following representation 2 A ( x) A ( x) = b x. 0 Theorem 2 allows us to obtain a recurrence relation for the a coefficients using the Ordered Bell Numbers. Theorem 3. For every 1 we have (6) a 0 = 1, a = 1 ( 1) j 1 2b j a j. Finally, from a theorem of E. A. Bender [1, Theorem 2] (quoted in [4, Theorem 7.3]), we give an asymptotic estimation for the a s. Theorem 4. We have the following asymptotic approximation ( ) (7) a = ( 1) 1 ( 1)! 1 log +1 1 + O 2 as +. The table below shows the exact and the approximate values (rounded to the nearest integer) given by (7) of the a coefficients for some. Relative errors are displayed as well. exact approximation relative error 1 2 2 0.00 2-1 -3 2.00 5 138 216 0.57 10-17070418 -20449513 0.20 15 27552372478592 30704425463714 0.11 20-247772400254700937149 -267768779624438420309 0.08 Table 1. The exact and the estimated values of the a coefficients for some values of. 3. THE PROOFS OF THE THEOREMS
4 G. Nemes Proof of Theorem 1. A formal manipulation gives x A = x a = a x x j j 0 0 j 0 = j x = ( 1) j 1 0 0 Let A,j := ( 1) j 1, then we find () 2 x A a 0 (2a 0 + a 1 ) x = = 1 2 A,j + 2 A 1,j + A 2,j x 2 = 2 a ( 3) a 1 + (A,j + 2A 1,j + A 2,j ) x 2 = 2 a ( 3) a 1 + ( 1) j 3 x 2 = ( 1) j 3 x. 2 Hence (8) () 2 A ( x ) On the other hand we have = 0 (9) A 2 (x) = a x 0 2 = 0 ( 1) j 3 a j x. Thus from (8) and (9) by (3) it follows that a j = ( 1) j 3, and hence 1 ( a = ( 1) j 3 ) a j, x. x.
On the coefficients of an asymptotic expansion 5 for 3. Proof of Theorem 2. Simple formal manipulation gives log A (x) = log (1 + jx) 1/2j = 1 log (1 + jx) 2j j 1 j 1 = 1 ( 1) 1 j 2 j x = j ( 1) 1 2 2 j+1 x j 1 1 1 j 1 = 1 ( 1) 1 2b x. Proof of Corollary 1. From (5) we find 2 A ( x) A ( x) = 2 1 ( 1) 1 2b ( x) 1 = 1 + b x = b x. 1 0 Proof of Theorem 3. From (5) we have exp ( 1) 1 2b x = a x. 1 0 By differentiating each side with respect to x, we obtain ( 1) 1 2b x 1 exp ( 1) 1 2b x = a x 1, 1 1 0 ( 1) 2b +1 x a x = ( + 1) a +1 x, 0 0 0 ( 1) j 2b j+1 a j x = ( + 1) a +1 x. 0 0 Equating the coefficients in both sides gives (6). Theorem 5 (E. A. Bender). Suppose that α (x) = α x, F (x, y) = f h x h y, 1 h, 0 β (x) = 0 β x = F (x, α (x)), D (x) = 0 δ x = F (x, y) y. y=α(x) Assume that F (x, y) is analytic in x and y in a neighborhood of (0, 0), α 0 and
6 G. Nemes Then (1) (2) α 1 = o (α ) as +, r α j α j = O (α r ) for some r > 0 as +. j=r r 1 β = δ j α j + O (α r ) as +. Proof of Theorem 4. We apply Theorem 5 to the functions α (x) := 1 ( 1) 1 2b x, F (x, y) := e y. It follows that β (x) = D (x) = A (x) = 0 a x. H. S. Wilf [7, p. 190] showed that! (10) b = 2 log +1 2 ( ( 1 + O (0.16 log 2) )) =! 2 log +1 2 ( 1 + O ( 0.12 )) as + (a complete asymptotic expansion can be found in [3, p. 269]). Hence α := ( 1) 1 2b = ( 1) 1 ( 1)! ( ( 1 + O 0.12 )) log +1 2 and (11) c 1 ( 1)! log +1 2 < α < c 2 ( 1)! log +1 2 for some c 1 > c 2 > 0. Since α 0 and ( 1) ( 2)! ( ) 1 + O 0.12 1 α 1 log 2 lim = lim + α + ( 1) 1 ( 1)! log +1 2 (1 + O (0.12 )) log 2 ( ( = lim 1 + O 0.12 1 )) = 0, + 1
On the coefficients of an asymptotic expansion 7 the condition (1) of Theorem 5 holds. From (11) 1 α j α j < c 2 2 1 (j 1)! log j+1 2 = c2 2 2 ( 2)! log 2 2 log 2 ( 1)! log j+1 2 = c2 1 2 ( 2)! (j 1)! ( 1)! log 2 2 log 2 ( 2)! ( 2)! 1 ( 2 j ) = O log 2 = O (α 1 ). Hence condition (2) holds with r = 1. Since F (x, y) = e y is analytic in x and y, it follows that a = a 0 α + O (α 1 ) = ( 1) 1 ( 1)! ( ( 1 + O 0.12 )) ( 2)! log +1 + O 2 log 2 ( ) = ( 1) 1 ( 1)! 1 log +1 1 + O 2 as +. Acnowledgment. I would lie to than Antal Nemes and the two anonymous referees for their thorough, constructive and helpful comments and suggestions on the manuscript. REFERENCES 1. E. A. Bender: An asymptotic expansion for the coefficients of some formal power series. J. London Math. Soc. 9 (1975), 451 458. 2. S. R. Finch, Mathematical Constants, Cambridge University Press, 2003. 3. P. Flajolet, R. Sedgewic, Analytic Combinatorics, Cambridge University Press, 2009. 4. A. M. Odlyzo, Asymptotic enumeration methods, in Handboo of Combinatorics (Vol. II) (eds.: R. L. Graham, M. Grötschel and L. Lovász), M.I.T. Press and North- Holland, 1995, 1063 1229. 5. M. Somos: Several Constants Related to Quadratic Recurrences. Unpublished note, 1999. 6. J. Sondow, P. Hadjicostas: The generalized-euler-constant function γ (z) and a generalization of Somos s quadratic recurrence constant. J. Math. Anal. Appl. 332 (2007), 292 314. 7. H. S. Wilf, Generatingfunctionology, 3rd. ed., A K Peters, Ltd., 2006. Loránd Eötvös University H-1117 Budapest, Pázmány Péter sétány 1/C Hungary E-mail: nemesgery@gmail.com