Benaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco

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1 EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco Mustapha Rachidi Départemet de Mathématiques et Iformatique, Faculté des Scieces, Uiversité Mohammed V, B.P. 1014, Rabat, Morocco Osamu Saeki Faculty of Mathematics, Kyushu Uiversity, Hakozaki, Fukuoka , Japa (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 ad the liear recurrece of order r = a 0 + a 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 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

2 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

3 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)

4 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 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 k r 1 )! a k 0 0 k 0!k 1!... k r 1! ak 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 k r 1 1)! k 0!k 1!... k j 1!(k j 1)!k j+1!... k r 1! = (k 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

5 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 k r 1 )! a k 0 0 k 0!k 1!... k r 1! ak ak r 1 r 1 (k k r 1 1)! k 0!... (k j 1)!... k r 1! ak ak j 1 (k k r 1 1)! k 0!... (k j 1)!... k r 1! ak ak j j... a k r 1 r 1 (k 0 + k k r 1 )! a k 0 0 k 0!k 1!... k r 1! ak 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

6 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

7 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

8 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 k 1 )! a k 0 0 k 0!k 1!... k 1! ak 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 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

9 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 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

10 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

11 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 ), 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, [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): [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): [4] C. Breziski ad M. Redivo Zaglia. Extrapolatio Methods. Theory ad Practice. Studies i Computatioal Math. 2, North-Hollad Publishig Co., Amsterdam, [5] W.Y.C. Che ad J.D. Louck. The Combiatorial Power of the Compaio Matrix. Liear Algebra Appl. 232 (1996): [6] F. Dubeau, W. Motta, M. Rachidi ad O. Saeki. O Weighted r-geeralized Fiboacci Sequeces. The Fiboacci Quarterly 35 (1997): [7] L. Euler. Itroductio to the Aalysis of the Ifiite, Book 1. Spriger-Verlag, [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., [10] J.A. Jeske. Liear Recurrece Relatios, Part I. The Fiboacci Quarterly 1 (1963): [11] W.G. Kelley ad A.C. Peterso. Differece Equatios. A Itroductio with Applicatios. Academic Press, Ic., Bosto, MA, [12] C. Levesque. O m-th Order Liear Recurreces. The Fiboacci Quarterly 23 (1985): [13] E.P. Miles. Geeralized Fiboacci Sequeces by Matrix Methods. The Fiboacci Quarterly 20 (1960): [14] W. Motta, M. Rachidi ad O. Saeki. O -Geeralized Fiboacci Sequeces. The Fiboacci Quarterly 37 (1999): [15] W. Motta, M. Rachidi ad O. Saeki. Coverget -Geeralized Fiboacci Sequeces. The Fiboacci Quarterly 38 (2000): [16] M. Moulie ad M. Rachidi. Applicatio of Markov Chais Properties to r-geeralized Fiboacci Sequeces. The Fiboacci Quarterly 37 (1999): [17] M. Moulie ad M. Rachidi. -Geeralized Fiboacci Sequeces ad Markov Chais. The Fiboacci Quarterly 38 (2000): AMS Classificatio Numbers: 40A05, 30C15, 40A25, 41A60 65