The asymptotic behavior of the real roots of Fibonacci-like polynomials

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1 Act Acdemie Pedgogice Agriensis, Sectio Mthemtice, ) pp The symptotic behvior of the rel roots of Fiboncci-like polynomils FERENC MÁTYÁS Abstrct. The Fiboncci-like polynomils G n x) re defined by the recursive formul G n x)xg n x)+g n x) for n, where G 0 x) nd G x) re given seedpolynomils. In this pper the non-zero ccumultion points of the set of the rel roots of Fiboncci-like polynomils re determined if either both of the seed-polynomils re constnts or G 0 x) nd G x)x± R\{0}). The theorems generlize the results of G. A. Moore nd H. Prodinger who investigted this problem if G 0 x) nd G x)x, furthermore we extend result of Hongqun Yu, Yi Wng nd Mingfeng He. Introduction The Fiboncci-like polynomils G n x) re defined by the following mnner. For n ) G n x) xg n x) + G n x), where G 0 x) nd G x) re fixed polynomils so-clled seed-polynomils) with rel coefficients. If it is necessry to denote the seed-polynomils, then we will use the nottion G n x) G n G 0 x),g x),x), too. The polynomils G n 0,,x) re the originl Fiboncci polynomils nd the numbers G n 0,,) re the well-known Fiboncci numbers. Recently, G. A. Moore [5] investigted the mximl rel roots g n of the polynomils G n,x,x) nd proved tht g n exists for every n nd lim n g n 3/. These numbers g n re clled s golden numbers. ) H. Prodinger [6] gve the symptotic formul g n )n 4 n. Hongqun Yu, Yi Wng nd Mingfeng He [3] investigted the limit of the mximl rel roots g n of polynomils G n,x,x) if R +. For brevity let us introduce the following nottions. B denotes the set of the rel roots of polynomils G n x) n 0,,,...) nd A denotes the set of the the ccumultion points of set B. In [4] we investigted these sets. Reserch supported by the Hungrin Ntionl Reserch Science Foundtion, Operting Grnt Number OTKA T 0095.

2 56 Ferenc Mátyás Although, the min result of [4] is formulted for seed-polynomils with integer coefficients but it is true for seed-polynomils with rel coefficients, too. Since we re going to pply it, therefore we cite it s lemm. Lemm. Let G 0 x) nd G x) be two fixed polynomils with rel coefficients, G 0 0) G 0) 0 nd x 0 R. x 0 A if nd only if one of the following conditions holds: i) G x 0 ) G 0 x 0 ) αx 0 ) nd x 0 > 0; ii) G x 0 ) G 0 x 0 ) βx 0 ) nd x 0 < 0; iii) x 0 0, where ) αx) x + x + 4 nd βx) x x + 4. The purpose of this pper is to investigte the symptotic behvior of the elements of the set B in the cses of simple seed-polynomils. In our discussion we re going to use the following explicit formule for the polynomil G n x) G n G 0 x),g x),x). It is known tht 3) G n x) px)α n x) qx)β n x) for n 0, where αx) nd βx) re defined in ), while px) G x) βx)g 0 x) αx) βx) nd qx) G x) αx)g 0 x). αx) βx) These formule cn be obtined by stndrd methods or see in []. Since we wnt to investigte the roots of the polynomils G n x), therefore it is worth rephsing the expression G n x) 0 s px) qx) ) n βx), αx) tht is 4) G x) βx)g 0 x) G x) αx)g 0 x) x ) n x + 4 x +. x + 4 Let us consider the polynomil G n G 0 x),g x),x). It is obvious tht G n 0,0,x) is identicl to the zero polynomil for every n 0. Using 3)

3 The symptotic behvior of the rel roots of Fiboncci-like polynomils 57 the identities G n 0,G x),x) G x) G n 0,,x) nd G n G 0 x),0,x) G 0 x) G n,0,x) yield. But it is known from [] nd cn be obtined esily from 4) tht neither the Fiboncci polynomils G n 0,,x) nor the polynomils G n,0,x) hve rel root x except x 0 if n is even or odd, respectively. Therefore investigting the symptotic behvior of the roots of polynomils G n G 0 x),g x),x) we cn ssume tht the seedpolynomils differ from the zero polynomil nd t lest one of them is monic polynomil since one cn simplify the left-hnd side of 4) with the leding coefficient of the polynomil G x) or G 0 x)). Theorems nd Proofs First of ll we need the following lemm, which dels with the properties of the functions αx) nd βx) defined in ). Lemm. ) On the intervl [0, ) the function αx) is continuous nd strictly monotoniclly decresing, its grph is convex nd b) On the intervl,0] the function βx) αx) > 0. is continuous nd strictly monotoniclly decresing, its grph is concve nd 0 > βx). Proof. By ) it is obvious tht the functions αx) nd re continuous on the bove mentioned intervls. The rest of the sttement cn be βx) proved esily using the methods of differentil clculus. Further on we del with the set A if G 0 x) nd G x). In this cse, using Lemm, the set A cn be determined in very simple mnner. Theorem. Let R \ {0} nd { G n,,x) } be Fiboncci-like polinomils. If 0 < < then A \ {0}, while in the cse A \ {0}. Proof. According to Lemm to get the elements of the set A \ {0} we hve to solve the equtions 5) nd x + x + 4 for x > 0 6) x x + 4 for x < 0.

4 58 Ferenc Mátyás By Lemm the functions αx) x+ nd x +4 βx) x re continuous, > x +4 > 0 for ny x > 0 nd 0 > > for ny x < 0, αx) βx) therefore 0 < < is necessry nd sufficient condition for the solvbility of 5) nd 6). Solving 5) nd 6) we get tht the single rel root x 0 is x 0, where x 0 > 0 if < < 0 nd x 0 < 0 if 0 < <. This completes the proof. In the following theorems we prove symptotic formule for those rel roots g n of the polynomils G n,x ±,x) which do not tend to 0 if n tends to infinity. Theorem. Let G 0 x) nd G x) x, } where R \ {0}. If either > 0 or < then A \ {0}, while in the cse { +) + < 0 we hve A \ {0}. Furthermore for lrge n g n + ) + + )n + + ) + ) + ) + ) n. Proof. According to Lemm, x 0 A \ {0} if nd only if 7) or 8) x 0 x 0 x 0 + x nd x 0 > 0 x 0 x nd x 0 < 0 holds. Using the sttements of Lemm one cn verify tht 7) hs solution for x 0 if nd only if > 0, while 8) hs solution for x 0 if nd only if <. Solving 7) nd 8) we get tht x 0 + ) +. To determine the symptotic behvior of g n we pply 4), which in our cse hs the following form g n ) + g n ) gn + 4 g n ) + g n + ) gn + 4 g n ) n gn + 4 g n +. gn + 4

5 The symptotic behvior of the rel roots of Fiboncci-like polynomils 59 This will be much nicer when we substitute 9) g n u u. Without loss of generlity we cn ssume tht u > 0 nd we get the equlity 0) u + u + )u ) + u)u + ) u ) n. Since x 0 u u holds for u + nd u + therefore it is plin to see tht, for lrge n, 9) cn only hold if u is either close to + or +. In both cses this would men tht g n is close to x 0. Let us ssume tht u is close to + nd so > 0 becuse of u > 0. It is cler from 0) tht the cses when n is even or odd hve to be distinguished. We strt with n m nd rewrite 0) s u + u + )u ) ) + u u 4m. u + We get the symptotic behvior by process known s bootstrpping which is explined in []. First we insert u + + δ into the left-hnd side of ) nd u + into the right-hnd side of ). So we get n pproximtion for δ. Then we insert u + + δ + δ into the left-hnd side of ) nd u + + δ into the rihgt-hnd side of ) nd get n pproximtion for δ. This procedure cn be repeted to get better nd better estimtions for u. Now we determine only the number δ. From ) we hve δ + + ) + ) 4m + nd so u + + δ ) + ) 4m. + Substituting u into 9) we get tht ) g m ++δ + + δ + ) ) + ) + ) +) 4m. If n m + then 0) cn be rewrite s + u u + u + )u ) u u 4m. u + )

6 60 Ferenc Mátyás Using the bootstrpping method for u + + δ we get the estimtion which implies the following form: δ + + ) + ) + ) + ) 4m, 3) g m+ + + δ + + δ + ) ) + ) + ) 4 + ) 4m. Compring ) nd 3) the desired pproximtion yields since > 0. One cn verify in the sme mnner tht the estimtion for g n lso holds when <. This completes the proof. Remrk. From our proof one cn see tht for lrge n g n g n if > 0 while g n is the miniml rel root if <. A similr result cn be proved for the polynomils G n,x +,x). Theorem 3. Let G 0 x) nd G {x) x + } where R \ {0}. If etiher > 0 or < then A \ {0} +) +, while A \ {0} if < 0. Furthermore for lrge n + ) g n + + )n + + ) + ) + ) + ) n, where G n g n ) 0 nd lim n g n 0. Proof. For rel number x 0, by our Lemm, x 0 A \ {0} if nd only if 4) or 5) x 0 + x 0 + x 0 + x nd x 0 > 0 x 0 x nd x 0 < 0 holds. Substituting x 0 for x 0 into 4) nd 5) we get tht 6) x 0 x 0 x nd x 0 < 0

7 nd 7) The symptotic behvior of the rel roots of Fiboncci-like polynomils 6 x 0 x 0 + x nd x 0 > 0 Since 6) nd 7) re identicl to 8) nd 7), respectively, therefore ll of the sttements of our theorem follows from the Theorem. Thus the theorem is proved. Concluding Remrks Using our Theorem for we get tht g n g n )n 4 n, which mtches perfectly with the result of H. Prodinger. On the other hnd it is quite likely tht similr results cn be obtined for seed-polynomils G 0 x) x± nd G 0 x) or for other polynomils. This could be the subject of further reserch work. References [] D. Greene nd D. Knuth, Mthemtics for the Anlysis of Algorithms, Birkhäuser, 98. [] V. E. Hoggt, Jr. nd M. Bicknell, Roots of Fiboncci Polynomils, The Fiboncci Qurterly.3 973), [3] Hongqun Yu, Yi Wng nd Mingfeng He, On the Limit of Generlized Golden Numbers, The Fiboncci Qurterly ), [4] F. Mátyás, Rel Roots of Fiboncci-like Polynomils, Proceedings of Number Theory Conference, Eger 996) to pper) [5] G. A. Moore, The Limit of the Golden Numbers is 3/, The Fiboncci Qurterly ), 7. [6] H. Prodinger, The Asymptotic Behvior of the Golden Numbers, The Fiboncci Qurterly ), 4 5. Ferenc Mátyás Eszterházy Károly Techers Trining College Deprtment of Mthemtics Leányk u Eger, Pf. 43. Hungry E-mil: mtys@gemini.ektf.hu