EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha Rachidi Départemet de Mathématiques et Iformatique, Faculté des Scieces, Uiversité Mohammed V, B.P. 1014, Rabat, Morocco e-mail: rachidi@fsr.ac.ma Osamu Saeki Faculty of Mathematics, Kyushu Uiversity, Hakozaki, Fukuoka 812-8581, Japa e-mail: saeki@math.kyushu-u.ac.jp (Submitted July 2001-Fial Revisio May 2002) 1. INTRODUCTION Let a 0, a 1,..., a r 1 (a r 1 0) ad α 0, α 1,..., α r 1 (r 1) be two sequeces of real or complex umbers. The sequece { } r+1 defied by = α for r + 1 0 ad the liear recurrece of order r = a 0 + a 1 1 + + a r 1 r+1 ( 0) (1.1) is called a weighted r-geeralized Fiboacci sequece. Such sequeces have bee extesively studied i the literature (see [6, 10, 11, 13] for example). I this paper we shall refer to such a object as a sequece of type (1.1). Such sequeces have iterested may authors because of their various applicatios. For example, i umerical aalysis some discretizatio by fiite divisios gives such a liear recurrece relatio (for example, see [2, 4, 8, 9]). Sequeces of type (1.1) have bee geeralized i [14, 15] as follows. Let {a j } j 0 ad {α j } j 0 be two sequeces of real or complex umbers. The sequece {V j } j Z defied by = α ( 0) ad the liear recurrece of order V = a 0 + a 1 1 + + a m m +... ( 0) (1.2) is called a -geeralized Fiboacci sequece. Such sequeces have bee studied uder some hypotheses o the two sequeces {a j } j 0 ad {α j } j 0 which guaratee the existece of the terms for every 1 (see [3, 14, 15, 17]). The origi of r- or -geeralized Fiboacci sequeces goes back to Euler. I [7, Chapter XVII] he discussed Daiel Beroulli s method of usig liear recurreces to approximate zeros of (maily polyomial) fuctios. I this paper, we first study the relatioship betwee a give polyomial fuctio ad the associated sequece of type (1.1), ad the we use it to approximate ad fid a zero of the polyomial through Beroulli s method ( 2). Our results will be a bit weaker tha the usual oes; evertheless, we have icluded them i the aim to geeralize them to the case of geeral holomorphic fuctios. I 3 ad 4, this will be carried out through the use of -geeralized Fiboacci sequeces. These results are very importat, sice, as far as the authors kow, there has bee practically o method for approximatig or fidig a zero of a arbitrary holomorphic 55
fuctio usig the coefficiets i their power series expasios. Furthermore, i 4, we will discuss the approximatio process by usig r-geeralized Fiboacci sequeces with r fiite (see [3]), which will eable us to obtai more precise results. 2. BERNOULLI S METHOD FOR POLYNOMIAL FUNCTIONS I order to approximate a root of a polyomial P r (X) of degree r, Beroulli cosidered a sequece { } r+1 of type (1.1) such that P r (X) is its characteristic polyomial. More precisely, he used the iitial values 0 = 1 ad 1 = = r+1 = 0. It is well kow that uder certai coditios, if q exists, the it is a root of P r (X) such that q q for ay other root q of P r (X) (see [8, 9] or [6, Theorem 7], for example). The aim of this sectio is to establish similar results by usig the theory of holomorphic fuctios. Let Q r (z) = 1 a 0 z a r 2 z r 1 a r 1 z r be a complex polyomial of degree r(r 1, a r 1 0), ad cosider the complex fuctio f r (z) = 1/Q r (z). Sice Q r (0) = 1 0, the Taylor expasio of f r (z) i a disk cetred at 0 ca be writte as f r (z) = z (2.1) for some complex umbers 0, 1,.... The idetity Q r (z)f(z) = 1 implies that for all 0, where 0 = 1 ad 1 r 1 = a j j = = r+1 = 0. Hece, { } r+1 is a sequece of type (1.1) ad its characteristic polyomial coicides with P r (X) = X r a 0 X r 1 a r 2 X a r 1. Remark 2.1: Coversely, suppose that { } r+1 is a sequece of type (1.1) such that 0 = 1 ad 1 = = r+1 = 0. The we have f r (z) = where Q r (z) = 1 a 0 z a r 2 z r 1 a r 1 z r. z = 1 Q r (z), The polyomial fuctio Q r has a root ad Q r (0) 0. Hece, the fuctio f r = 1/Q r has a Taylor expasio ear 0 ad it is defied i the ope disk of radius R = mi{ λ ; λ is a root of Q r }. 56
Note that we always have 0 < R <. Thus, by usig the stadard theory of power series (for example, see [1]), we ca prove the followig (for more details, see the proof of Theorem 3.2 i the ext sectio). Propositio 2.2: Let Q r (z) = 1 a 0 z a r 2 z r 1 a r 1 z r (a r 1 0) be a complex polyomial of degree r. Cosider the sequece { } r+1 of type (1.1) whose coefficiets ad iitial values are give by a 0, a 1,..., a r 1 ad 0 = 1, 1 = = r+1 = 0 respectively. We suppose that 0 for all sufficietly large. The the radius of covergece R of the series (2.1) satisifies lim if R lim sup ad R = mi{ λ i λ is a root of Q r }. I particular, we have Q r (Re iθ ) = 0 for some θ [0, 2π), ad R µ for all other roots µ of Q r. As a immediate corollary, we have the followig. Corollary 2.3: I the above propositio, if Λ exists, the Λ is the smallest amog the moduli of the roots of Q r. Remark 2.4: As we oted before, if λ exists, the actually λ itself is a root of Q r with the smallest modulus (for example, see [6]). I fact, we ca easily show that Q r (λ ) = 0 as follows: Q r (λ ) 1 a 0 1 a 0 1 a 0 a 1 ( a 1 a 1 1 ) 2 a r 1 ( 1 ) r a r 1... a r 1 (r 1) a 0 a 1 1 a r 1 (r 1) 57 = 0. (r 1) (r 1)
Example 2.5: Cosider the usual Fiboacci sequece {F } 1, which is a sequece of type (1.1) with r = 2. I this case, the correspodig polyomial is Q 2 (z) = 1 z z 2. Furthermore, it is well kow that λ (2) F 1 = F 5 1. 2 It is easy to verify that λ is the root of Q 2 with the smallest modulus. Remark 2.6: I the above results, the coditio that 0 for all sufficietly large is essetial. For example, if r is eve ad Q r (z) is a polyomial of z 2, the i the power series expasio of f r (z), the coefficiets with odd are all zero. Thus we caot cosider / for eve. We have a combiatorial expressio for sequeces of type (1.1) as follows. Propositio 2.7: Let { } r+1 be a sequece of type (1.1) whose coefficiets ad iitial values are a 0, a 1,..., a r 1 ad 0 = 1, 1 = = r+1 = 0 respectively. The we have = k 0 +2k 1 + +rk r 1 = (k 0 + k 1 + + k r 1 )! a k 0 0 k 0!k 1!... k r 1! ak 1 1... ak r 1 r 1 (2.2) for all r + 1, where k 0, k 1,..., k r 1 ru over oegative itegers. Proof: Let us prove the assertio by iductio o. It is easy to see that it is true for 0. Suppose that 0 ad that the assertio is true for all itegers less tha or equal to. It is easy to see that r 1 (k 0 + k 1 + + k r 1 1)! k 0!k 1!... k j 1!(k j 1)!k j+1!... k r 1! = (k 0 + + k r 1 )! k 0!... k r 1! holds, where we igore the terms correspodig to those j with k j = 0. The, usig this, we 58
see that r 1 = a j j r 1 = r 1 = = = a j a j k 0 +2k 1 + +rk r 1 = j k 0 +2k 1 + +rk r 1 =,k j 1 r 1 k 0 +2k 1 + +rk r 1 = k 0 +2k 1 + +rk r 1 = This completes the proof. if (k 0 + k 1 + + k r 1 )! a k 0 0 k 0!k 1!... k r 1! ak 1 1... ak r 1 r 1 (k 0 + + k r 1 1)! k 0!... (k j 1)!... k r 1! ak 0 0... ak j 1 (k 0 + + k r 1 1)! k 0!... (k j 1)!... k r 1! ak 0 0... ak j j... a k r 1 r 1 (k 0 + k 1 + + k r 1 )! a k 0 0 k 0!k 1!... k r 1! ak 1 1... ak r 1 r 1. j... a k r 1 r 1 Compare the above propositio with [5, 12, 16]. Let us deote the right had side of the equatio (2.2) by ρ(, r). The by Corollary 2.3, Λ ρ(, r) ρ( + 1, r) exists, the (Λ ) 1 is the largest amog the moduli of the roots of the characteristic polyomial P r (X), ad the radius of covergece R of the Taylor series (2.1) of f r (z) = 1/Q r (z) coicides with Λ. Furthermore, if λ ρ(, r) ρ( + 1, r) exists, the λ is a root of Q r as we have see i Remark 2.4. I other words, we ca approximate a root of Q r with the smallest modulus by usig a 0, a 1,..., a r 1 together with the combiatorial formula (2.2). Remark 2.8: The Taylor expasio of the complex fuctio f r (z) = 1/Q r (z) i the ope disk D(0; R), with R beig as above, is give by f r (z) = 1! f () r (0)z. Thus, from the expressio (2.1) we derive that f r () (0) =! for all 0. 59
3. THE BERNOULLI-EULER METHOD FOR HOLOMORPHIC FUNCTIONS I this sectio, we show that Beroulli s method for approximatig ad fidig a root of a polyomial fuctio preseted i 2 ca be exteded to the case of holomorphic fuctios. Let Q(z) be a complex fuctio which is holomorphic i a eighbourhood of 0. Let R 1 > 0 be the largest positive umber such that Q is holomorphic i the ope disk D(0; R 1 ). I order to study the zeros of Q i D(0; R 1 ) {0}, we may oly cosider the case where Q takes the form Q(z) = 1 a j z j+1. (3.1) Sice Q(0) = 1 0, f(z) = 1/Q(z) has a Taylor expasio i a certai disk cetred at 0, which is of the form f(z) = z. (3.2) The idetity Q(z)f(z) = 1 implies that we have V = a j j for all 0, where V 0 = 1 ad V j = 0 ad for all j 1. Hece, { } Z is a -geeralized Fiboacci sequece as i (1.2) whose iitial values are give by V 0 = 1 ad V j = 0 for all j 1. Remark 3.1: Coversely, suppose that { } Z is a sequece as i (1.2) such that V 0 = 1 ad V j = 0 for all j 1. The, we have f(z) = formally, where Q(z) is give by (3.1). z = 1 Q(z) As a direct geeralizatio of Propositio 2.2, we have the followig. Theorem 3.2: Let Q(z) = 1 a j z j+1 be a holomorphic fuctio defied i a eighbourhood of the origi with radius of covergece R 1 > 0. Cosider the sequece { } Z as i (1.2) whose coefficiets ad iitial values are give by {a j } j 0 ad V 0 = 1, V j = 0 for all j 1, respectively. We suppose that 0 for all sufficietly large ad that the radius of covergece R of the series (3.2) satisfies R < R 1. The, we have lim if V R lim sup 60 V
ad R = mi{ λ ; λ is a zero of Q}. I particular, we have Q(Re iθ ) = 0 for some θ [0, 2π), ad R µ for all other zeros µ of Q. ( ) 1 Proof: It is well kow that R sup V (for example, see [1]). Let L be a arbitrary real umber such that 0 < L < lim if /V. The there exists a N such that /V > L for all N. Therefore, V N+k < V N L k for k = 1, 2, 3,..., ad hece N+k V N+k < N+k V N L k = L 1 N+k V N L N. This implies that R 1 = lim sup k N+k V N+k L 1. Sice L is arbitrary, we coclude that lim if /V R. By a similar argumet, we ca show that R lim sup /V. For the secod part, first ote that Q(z) has o zero i the ope disk z < R, sice otherwise the radius of covergece R of f(z) = 1/Q(z) would be strictly smaller tha R. Suppose that Q(z) has o zero o the circle z = R. The it has o zero i the ope disk D(0, R + ε) for some ε > 0 (recall that R < R 1 ). It follows that the radius of covergece R of f(z) = 1/Q(z) is strictly greater tha R, which is a cotradictio. Therefore, we have R = mi{ λ ; Q(λ) = 0} ad we have Q(Re iθ ) = 0 for some θ [0, 2π). As a immediate corollary, we have the followig. Corollary 3.3: I the above theorem, if Λ V exists ad Λ < R 1, the Λ is the smallest amog the moduli of the zeros of Q. Remark 3.4: Eve if we assume that 0 for all, we do ot have R if or R sup i geeral. For example, set V V ad g(z) = 1 + z 2 + z 4 + z 6 +, h(z) = z + z(2z) 2 + z(2z) 4 + = zg(2z), f(z) = g(z) + h(z) = z. The radius of covergece of g is equal to 1, while that of h is equal to 1/2. Hece the radius of covergece of f is equal to R = 1/2. However, we have { 2, if is eve, = V 2 1, if is odd. 61
Thus we have lim sup V = +, So, either of them gives R i this example. Remark 3.5: Suppose that lim if V = 0. λ V exists. The we do ot kow if λ itself is a zero of Q with the smallest modulus. Compare this with Remark 2.4. I 4 we will give a partial aswer to this questio. Remark 3.6: I Corollary 3.3, if Λ R 1, the Q does ot have a zero i the ope disk D(0; R 1 ). By usig Propositio 2.7, we ca prove the followig combiatorial expressio for { } Z. Propositio 3.7: Let { } Z be a sequece as i (1.2) whose coefficiets ad iitial values are {a j } j 0 ad V 0 = 1, V j = 0 for all j 1, respectively. The we have for all Z. = ρ(, ) = By Corollary 3.3, if k 0 +2k 1 + +k 1 = Λ (k 0 + k 1 + + k 1 )! a k 0 0 k 0!k 1!... k 1! ak 1 1... ak 1 1 (3.3) ρ(, ) ρ( + 1, + 1) exists ad is strictly smaller tha R 1, the Λ is the smallest amog the moduli of the zeros of Q. Furthermore, the radius of covergece R of the Taylor series (3.2) of f(z) = 1/Q(z) coicides with Λ. We also have f () (0) =! for all 0 as i Remark 2.8. 4. THE BERNOULLI-EULER METHOD BY APPROXIMATION PROCESS I this sectio, we will use the results of 2 i order to approximate a zero of a holomorphic fuctio by usig r-geeralized Fiboacci sequeces with r fiite. The idea is very similar to that of [3]. Let Q(z) be a complex fuctio which is holomorphic i a eighbourhood of the origi. Let R 1 > 0 be the largest positive real umber such that Q is holomorphic i the ope disk D(0; R 1 ). As i the previous sectio, we suppose that its Taylor series expasio takes the form (3.1). Let { } Z be a -geeralized Fiboacci sequece as i (1.2) whose coefficiets ad iitial values are {a j } j 0 ad V 0 = 1, V j = 0 for all j 1, respectively. Note that exists for all Z. The followig approximatio has bee established i [3]: r (4.1) 62
for all 1, where for each r 1, the sequece { } r+1 is a type (1.1) defied by 0 = 1, = 0 for r + 1 1, ad = a 0 + + a r 1 r+1 for 0. However, i our case, (4.1) is trivial, sice we have = for r. Our first result of this sectio is the followig. Theorem 4.1: Let Q(z) = 1 a j z j+1 be a holomorphic fuctio defied i a eighbourhood of the origi with radius of covergece R 1 > 0. Cosider the doubly idexed sequece { } r+1,r 1 as above. We suppose the followig. (1) 0 for all sufficietly large ad r. (2) For all sufficietly large r, λ exists. (3) λ r λ exists ad we have λ < R 1. The λ is a zero of Q. Proof: Set Q r (z) = 1 a 0 z a r 2 z r 1 a r 1 z r. By Remark 2.4, we have ( ) ( ) lim Q r = Q r lim = Q r (λ ) = 0 for all sufficietly large r. Set T r (z) = Q(z) Q r (z). Note that for every R 1 with 0 < R 1 < R 1, we have lim r T r(z) = 0 uiformly for z R 1. We have Q(λ ) Q ( ) T r ( ) for all sufficietly large r. Hece we have Q(λ) r Q(λ ) r T r(λ ) = 0. This completes the proof. As a corollary, we have the followig. = T r (λ ) 63
Corollary 4.2: Let Q(z) = 1 a j z j+1 be a holomorphic fuctio defied i a eighbourhood of the origi with radius of covergece R 1 > 0. Cosider the doubly idexed sequece { } r+1,r 1 ad the sequece { } Z as above. We suppose the followig. (1), 0 for all sufficietly large ad r. (2) For all sufficietly large r, λ exists ad coverges uiformly with respect to r. (3) λ r λ exists ad we have λ < R 1. The we have ad it is a zero of Q. λ Proof: By our assumptios, we see that lim,r V = λ. The the result follows from (4.1) together with Theorem 4.1. Example 4.3: Let us cosider the example i [3, 7]. We shall use the same otatio. I this example, sice the coefficiets a i are all strictly positive real umbers, we have 0 for all 0 ad r 1. It has bee show that the sequeces { /qr } 1 are uiformly coverget for r 1 ad that lim q r = 1. Sice the sequece {q r } r 1 coverges to q > 0, the sequeces { / } 1 are also uiformly coverget ad coverge to qr 1 = p r for r 1. Furthermore, we have lim p r = p r ad 0 < p < R 1, where R 1 is the radius of covergece of Q (i [3, 7], R 1 is writte as R). Thus all the assumptios of Corollary 4.2 are satisfied ad p is a root of Q. 64
ACKNOWLEDGMENT The authors would like to express their sicere gratitude to the referee for may useful suggestios. The third author has bee partially supported by Grat-i-Aid for Scietific Research (No. 13640076), Miistry of Educatio, Sciece ad Culture, Japa. REFERENCES [1] L.V. Ahlfors. Complex Aalysis. A Itroductio to the Theory of Aalytic Fuctios of Oe Complex Variable. Third editio, Iteratioal Series i Pure ad Applied Math., McGraw-Hill Book Co., New York, 1978. [2] R. Be Taher ad M. Rachidi. Applicatio of the ε-algorithm to the Ratios of r- Geeralized Fiboacci Sequeces. The Fiboacci Quarterly 39 (2001): 22-26. [3] B. Beroussi, W. Motta, M. Rachidi ad O. Saeki. Approximatio of -Geeralized Fiboacci Sequeces ad Their Asymptotic Biet Formula. The Fiboacci Quarterly 39 (2001): 168-180. [4] C. Breziski ad M. Redivo Zaglia. Extrapolatio Methods. Theory ad Practice. Studies i Computatioal Math. 2, North-Hollad Publishig Co., Amsterdam, 1991. [5] W.Y.C. Che ad J.D. Louck. The Combiatorial Power of the Compaio Matrix. Liear Algebra Appl. 232 (1996): 261-278. [6] F. Dubeau, W. Motta, M. Rachidi ad O. Saeki. O Weighted r-geeralized Fiboacci Sequeces. The Fiboacci Quarterly 35 (1997): 102-110. [7] L. Euler. Itroductio to the Aalysis of the Ifiite, Book 1. Spriger-Verlag, 1988. [8] J. Gill ad G. Miller. Newto s Method ad Ratios of Fiboacci Numbers. The Fiboacci Quarterly 19 (1981): 1-4. [9] A.S. Householder. Priciples of Numerical Aalysis. McGraw-Hill Book Compay Ic., 1953. [10] J.A. Jeske. Liear Recurrece Relatios, Part I. The Fiboacci Quarterly 1 (1963): 69-74. [11] W.G. Kelley ad A.C. Peterso. Differece Equatios. A Itroductio with Applicatios. Academic Press, Ic., Bosto, MA, 1991. [12] C. Levesque. O m-th Order Liear Recurreces. The Fiboacci Quarterly 23 (1985): 290-295. [13] E.P. Miles. Geeralized Fiboacci Sequeces by Matrix Methods. The Fiboacci Quarterly 20 (1960): 73-76. [14] W. Motta, M. Rachidi ad O. Saeki. O -Geeralized Fiboacci Sequeces. The Fiboacci Quarterly 37 (1999): 223-232. [15] W. Motta, M. Rachidi ad O. Saeki. Coverget -Geeralized Fiboacci Sequeces. The Fiboacci Quarterly 38 (2000): 326-333. [16] M. Moulie ad M. Rachidi. Applicatio of Markov Chais Properties to r-geeralized Fiboacci Sequeces. The Fiboacci Quarterly 37 (1999): 34-38. [17] M. Moulie ad M. Rachidi. -Geeralized Fiboacci Sequeces ad Markov Chais. The Fiboacci Quarterly 38 (2000): 364-371. AMS Classificatio Numbers: 40A05, 30C15, 40A25, 41A60 65