Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1, F 1 = x 1 and for n 2, F n = F n 1 + x 2 F n 2. Denote the maxmum root of F n by g n. In ths artcle, we wll analyze the exstence and nonexstence of g n as well as study the behavor of the sequence {g n }. We wll prove all but one of the roots are rratonal and the sequence of the maxmum odd-ndexed roots are monotoncally ncreasng. 1 Introducton Consder the well-studed Fbonacc Polynomal Sequence, F n = xf n 1 + F n 2, for n 2 wth F 0 = 0, F 1 = 1. Many results are known about ths polynomal sequence. It s known that F n (1 s the n th Fbonacc number. Hogatt and Bcknell [HB also gave an explct form for the zeros of these polynomals. Further work ncludes Molna and Zeleke [MZ generalzng of the ntal condtons and explorng the recurson, F n = x k F n 1 + F n 2, now known as the Fbonacc-lke polynomals. They made a number of dscoveres about the asymptotc behavor of the roots of these polynomals. Ths nspred further work by Brandon Alberts n 2011. Alberts studed the Quas- Fbonacc polynomals. These are polynomals defned by the followng recurson: F q n = F q n 1 + x k F q n 2, for n 2 wth F q 0 = 1, F q 1 = x 1, where k = 1. He found a number of nterestng results ncludng the exstence of all roots and convergence of all roots to the same value, namely 2. He found that the roots of the even-ndexed polynomals converged from below and the roots of the odd-ndexed polynomals converged from above. In ths paper, we study a modfed recurson, a Quas-Fbonacc-Lke polynomal sequence, where k = 2. Specfcally, ths recurson s as follows: F q n = F q n 1 + x 2 F q n 2, for n 2 wth F q 0 = 1, F q 1 = x 1. 1
We wll explore the exstence and nonexstence of the roots, as well as ther behavor as a sequence. We wll also numercally and computatonally examne the asymptotc behavor of these roots n addton to showng all but one root are rratonal. We denote the maxmum root of F q n as g n, and for the sake of smplcty, we suppress the q superscrpt for the rest of ths paper. 2 Formulas and techncal results The formulas we wll use throughout ths paper are as follows: Lemma 1. F n (x = F 2n (x = [ n F 2n+1 (x = [( n 1 ( 2n x 2 n [( 2n x 2+1 ( n + [ n 1 x 2+1 ( 2n+1 x 2 (1 ( 2n 1 x 2+1 F n (x = (x 1 + λ λ n + (x 1 + λ + λ n λ + λ, where λ ± = 1 ± Remark. Formula (4 s known at the Bnet Form. 1 [( 0 = ( 1 + 1 (2 x 2 (3 1 + 4x 2. (4 2 F n (x = (1 + 2x 2 F n 2 (x x 4 F n 4 (x (5 Proof. Formula (1 We proceed by nducton. We begn by showng the base cases. Case 1: n = 1 [( ( 1 1 1 1 ( 1 + (x 1 x 2 Case 2: n = 2 ( 0 (x 1 0 x 0 + = [0 ( 1 + 1 (x 1 1 + 0 + 0 +... = x 1 = F 1 (x [( 2 1 1 [( 1 = ( 1 + 1 ( 1 + ( 2 1 ( 1 (x 1 0 [( 1 ( 1 + 0 (x 1 x 0 + x 2 [( 0 ( 1 + 0 ( 1 (x 1 x 2 +... 1 ( 0 (x 1 x 2 +... 1 = [0 ( 1 + 1 (x 1 1 + [1 ( 1 + 0 (x 1x 2 + 0 +... = x 2 + x 1 = F 2 (x 2
We now show the nductve step. Suppose ths dentty holds for all n < m. Then F m (x = F m 1 (x + x 2 F m 2 (x [( m 2 = ( 1 + 1 [( m 3 + x 2 1 [( m 2 = ( 1 + 1 [( m j 2 + j 2 j=1 Rendexng and combnng terms we have F m (x = = [ [(m 2 1 [( m 1 1 + ( m 2 2 ( 1 + ( m 1 ( m 2 (x 1 ( m 3 ( 1 + ( m 2 ( m j 2 ( 1 + (x 1 j 1 ( 1 + [( m 2 (x 1 x 2 gvng Formula (1. Formulas (2 and (3 follow drectly from Formula (1. x 2 (x 1 x 2 (x 1 + ( m 2 1 x 2 x 2j. (x 1 Formula (4: Ths s the Bnet Form for ths recurson. We proceed by usng the standard method of obtanng a Bnet Formula. We establsh the followng system of equatons: { c 1 λ 0 + + c 2 λ 0 = F 0 (x c 1 λ + + c 2 λ = F 1 (x where λ + and λ are the solutons to the equaton λ 2 = λ + x 2. From ths quadratc equaton, we fnd λ + and λ to be as n Formula (4. Solvng the system above, we fnd c 1 = x 1 + λ λ + λ and c 2 = (x 1 + λ + λ + λ. Formula (5: Ths follows from drect manpulaton of the recurson. Indeed we have x 2 F n (x = F n 1 (x + x 2 F n 2 (x = (1 + 2x 2 F n 2 x 2 F n 2 + x 2 F n 3 = (1 + 2x 2 F n 2 x 4 F n 4. Lemma 2. For all x < 0, we have 0 > F n (x > F n+1 (x for all n. 3
Proof. When x < 0, F 1 (x = x 1 < 1 = F 0 (x < 0. Furthermore, supposng ths statement holds up to F n 1, we see that snce F n 2 (x < 0, F n (x = F n 1 (x + x 2 F n 2 (x < F n 1 (x. Lemma 3. F n (0 = 1 for all n. Proof. F n (0 wll be the constant term of F n ; accordng to our Formula (1, ths wll be ( n 0 0 = 1 Lemma 4. lm x F 2n+1 (x = for all n. Proof. Frst we show ths s satsfed for ntal values of n. Proceedng by nducton we have the base cases lm F 1 = lm x 1 = x x lm F 3 = lm x 3 2x 2 + x 1 =. x x Suppose we ve shown ths for F 1, F 3,..., F 2n 1. By usng Formula (3, we have lm F 2n+1(x = lm x x n and [ ( 2n ( 1 + 2n (x 1 x 2. Snce the end behavor of a polynomal s determned by the sgn of the coeffcent on the hghest degree, t s suffcent to show that ths coeffcent s postve. Note that ( 2n = 0 when 2n >. However the leadng coeffcent must be nonzero, so 2n. Ths mples n. Examnng the term wth the hghest degree, we see [ ( ( 2n (n 2n (n + (x 1 (n 1 n [ ( ( n n = + (x 1 x 2n n 1 n [ ( n = + x 1 x 2n n 1 ( n = x 2n + x 2n+1 x 2n. n 1 x 2(n Thus the coeffcent of the term wth hghest degree, namely x 2n+1, s postve and therefore lm x F 2n+1 =. Lemma 5. lm x F 2n (x =, for all n. Proof. Lemma 5 can be proven usng smlar methods to Lemma 4. 4
3 Exstence and nonexstence of the roots Lemma 6. For any x and any n 1, 0 > F 2n (x F 2n+2 (x. In partcular, F 2n (x has no real roots. Proof. When x 0, ths statement follows drectly from Lemmas 2 and 3. So let x > 0. We now proceed by nducton, startng wth the base cases. Notce that F 2 (x = x 2 + x 1 < 0 and F 2 (x F 4 (x = x 4 2x 3 + 2x 2 = x 2 ((x 1 2 + 1 > 0, so 0 > F 2 (x > F 4 (x for all real values of x. We now show the nductve step. Suppose now that we have shown ths property through F 2n 2. Case 1: x 2. Then From Formula 5, we have 2x 2 F 2n 2 (x x 4 F 2n 2 (x x 4 F 2n 4 (x. F 2n (x = (1 + 2x 2 F 2n 2 (x x 4 F 2n 4 (x F 2n 2 (x. Case 2: x > 2 > 0. Recall c 1 = x 1 + λ λ + λ and c 2 = (x 1 + λ + λ + λ. 0 < x 0 > 4x 4x 2 4x + 1 < 1 + 4x 2 (2x 1 2 < 1 + 4x 2 Takng the prmary root of both sdes provdes, (2x 1 1 + 4x 2 < 0 c 1 < 0. A smlar proof shows c 2 < 0. Now we wll show 1 λ + x 2 < 0. 2 < x 0 < x 2 2 0 < 4x 4 8x 2 1 + 4x 2 < 4x 4 4x 2 + 1 Takng the prmary root of both sdes provdes A smlar proof shows 1 λ x 2 < 0. 1 2x 2 + 1 + 4x 2 < 0 1 λ + x 2 < 0. 5
Snce c 1, c 2, (1 λ + x 2, and (1 λ x 2 are all negatve, from Formula 4 gvng the Bnet form of F n, we have F 2n 2 F 2n = c 1 λ+ 2n 2 + c 2 λ 2n 2 c 1 λ 2n + c 2 λ 2n = c 1 λ+ 2n 2 (1 λ + x 2 + c 2 λ 2n 2 (1 λ x 2 > 0 Thus F 2n 2 > F 2n so that for all x, F 2n (x s negatve and therefore has no real roots. Lemma 7. F 2n+1 has at least one real root. Proof. Notce that the leadng coeffcent of F 2n+1 s postve and F 2n+1 (0 = 1. By the ntermedate value theorem, F 2n+1 has at least one real root. 4 Man Results For the remander of ths paper, let g n denote the maxmum real root of F n. Theorem 1. 1 = g 1 < g 3 < < g 2n+1 for all n. Proof. We proceed by nducton. Startng wth the base cases drect computaton shows 1 = g 1 < g 3. We now show the nducton step. Suppose we have shown ths through g 2n 1. Then snce g 2n 3 s a maxmum root and lm x F 2n 3 (x = +, we note that F 2n 3 (g 2n 1 > 0. So F 2n+1 (g 2n 1 = (1 + 2g 2 2n 1 F 2n 1 (g 2n 1 g 4 2n 1 F 2n 3 (g 2n 1 = g 4 2n 1 F 2n 3 (g 2n 1 < 0. Thus, snce lm x F 2n+1 (x = +, F 2n+1 must have a root g 2n+1 > g 2n 1. Theorem 2. The roots of F 2n+1 are rratonal for n > 0. Proof. If there s some ratonal number r whch s a root of F 2n+1 then, by the Ratonal Root Theorem and Formula 3, [ ( ( 2n n 2n n + n 1 n r = ± [ ( ( 2n 2n + 1 0 = ±( n + 1. Case 1: r = n + 1. By Theorem 1, g 2n+1 > 1. However for all n, we have r 1. So n + 1 cannot be a root. 6
Case 2: r = n 1. We have shown ths cannot be a root for n < 62 by drect computaton. Through manpulaton of Formula (3 we have n+1 [ ( ( 2n 2n F 2n+1 (x = + (x 1 x 2. 1 When substtutng r nto the expresson above for x, t s easy to show that only the coeffcent of the last term s negatve. It s gven by ( 2(n 1 2n. Fndng the rato between the orgnal bnomal coeffcents provdes, ( ( 2n 2n = 1 2n 2 + 1. Snce the last term s the only negatve term, t s suffcent to show that the thrd to last term s larger. Usng = n 2 wth n 62 gves, ( ( ( ( n + 2 n + 1 n 4n n 1 n 1 n 1 5 8 48. 5 ( ( (n 2(n + 1(n(n 1 4n 24 5 8 2(n 1 4 5 ( ( 2n (n 2 (n 2 n 2 2n 2(n 2 + 1 + n 2 (n 1 2(n 2 2(n 1 2n where the left hand sde s the thrd to last term n the above summaton. Therefore, g 2n+1 s rratonal for all n > 0. 5 Numercal evdence Our research has also suggested certan other results. One such possblty s that the oddndexed polynomals, F 2n+1, have exactly one real root. As the followng graph shows, the frst few odd-ndexed polynomals have exactly one real root. 7
Thus we have constructed the followng lemma and conjecture. Lemma 8. F 2n+1 has no real roots on (, g 2n 1 (g 2n+1,. Proof. Ths s obvous for F 1 (x = x 1. Suppose we have shown t through F 2n 1. When x > g 2n+1, ths follows drectly from maxmalty of g 2n+1. Therefore, let x g 2n 1. Recall that F 2n (x < 0 and note that F 2n 1 (x 0. Then F 2n+1 (x = F 2n (x + x 2 F 2n 1 (x < 0 The prevous lemma does not exclude the possblty of a root exstng on the nterval (g 2n 1, g 2n+1, provdng the followng conjecture formed through observaton. Conjecture 1. F 2n+1 has no real roots on (g 2n 1, g 2n+1. Ths has been confrmed by drect calculaton through F 599, meanng all of these polynomals have exactly one real root. It s our hope that future work may be able to formally prove ths for all odd-ndexed F n. Our work has also led us to beleve the followng conjecture: Conjecture 2. The sequence {g 2n+1 } s unbounded. The roots do not appear to asymptotcally approach any number through g 599, selected roots are shown below. 8
n g n 1 1 3 1.755 5 2.402 13 4.616 29 8.390 101 22.544 233 44.969 419 73.762 599 100.05 Graphcally, we can also see that they do not appear to asymptotcally approach any number. After multple regresson analyses, we can say that t appears to grow slghtly faster than logarthmcally. 6 Acknowledgements We would lke to thank Mchgan State Unversty and the Lyman Brggs College for ther support of our REU, as well as our faculty mentor, Dr. Akllu Zeleke. We would also lke to thank hs graduate assstants, Justn Droba, Ran Satyam and Rchard Shadrach. Project sponsored by the Natonal Securty Agency under Grant Number H98230-13-1-0259. Project sponsored by the Natonal Scence Foundaton under Grant Number DMS 1062817. Specal thanks to Danel Thompson. 9
7 References Brandon Alberts, On the Propertes of a Quas-Fbonacc Polynomal Sequence, preprnt, SURIEM 2011 V. E. Hoggatt, Jr. and Marjore Bcknell, Roots of Fbonacc polynomals, Fbonacc Quart., 11(3:271-274, 1973. Robert Molna and Akllu Zeleke, Generalzng results on the convergence of the maxmum roots of Fbonacc type polynomals, In Proceedngs of the Forteth Southeastern Internatonal Conference on Combnatorcs, Graph Theory and Computng, volume 195, pages 95-104, 2009. 10