Let $\left \{ a_{i}\right \}, \left \{ b_{i}\right \}$ be real numbers for $0\leqslant i\leqslant r-1$, and define a \textit{$r$-periodic sequence}$\left\{v_{n}\right \}$ with initial conditions $v_{0}$, $v_{1}$ and recurrences $v_{n}=a_{t}v_{n-1}+b_{t}v_{n-2}$ where $n \equiv t \left( \text{mod } r\right)$ ($n\geqslant 2$).In this paper, by aid of Chebyshev polynomials, we introduce a new method to obtain the complex factorization ofthe sequence $\left\{v_{n}\right \}$ so that we extend some recentresults and solve some open problems. Also, we provide new results by obtainingthe binomial sum for the sequence $\left\{v_{n}\right \}$ by using Chebyshev polynomials

Sahin, Murat

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXXIII (2018) – Fasc. I, pp. 179–189doi: 10.4418/2018.73.1.13COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALSMURAT SAHIN - ELIF TAN - SEMIH YILMAZLet {ai} ,{bi} be real numbers for 0 6 i 6 r− 1, and define a r-periodic sequence {vn} with initial conditions v0, v1 and recurrences vn =atvn−1 + btvn−2 where n ≡ t (mod r) (n > 2). In this paper, by aid ofChebyshev polynomials, we introduce a new method to obtain the com-plex factorization of the sequence {vn} so that we extend some recentresults and solve some open problems. Also, we provide new results byobtaining the binomial sum for the sequence {vn} by using Chebyshevpolynomials.Let {ai} and {bi} be real numbers for 0 6 i 6 r−1, and define a sequence{vn} with initial conditions v0, v1, and for n> 2,vn =a0vn−1+b0vn−2, if n≡ 0(mod r) ,a1vn−1+b1vn−2, if n≡ 1(mod r) ,......ar−1vn−1+br−1vn−2, if n≡ r−1(mod r) .(1)We call {vn} as a r-periodic sequences. It is studied in [7] by Panario et. al. andthey find the generating function and Binet’s like formula for the sequence {vn}via generalized continuant. Petronilho obtain the same Binet’s like formula byusing tools from ortogonal polynomials in [8].Entrato in redazione: 1 gennaio 2007AMS 2010 Subject Classification: 33C47,65Q30,11B39,05A15Keywords: Chebyshev polynomials, complex factorization, binomial sums180 MURAT SAHIN - ELIF TAN - SEMIH YILMAZFor r = 2 and initial values v0 = 1, v1 = a1, Cooper and Parry [3] calledthe sequence {vn} as the period two second order linear recurrence system, andgave the complex factorization of odd terms of this sequence by determining theeigenvalues and eigenvectors of certain tridiagonal matrices. The problems re-mained unsolved in [3] are determining the complex factorization of even termsof the period two second order linear recurrence system and determining thecomplex factorization of the sequence {vn} for given general r. Also, for r = 2and initial values v0 = 0, v1 = 1, Jun [5] give a connection between the sequence{vn} and Chebyshev polynomials of the second kind {Un(x)}. By using the fac-torization of {Un(x)}, Jun derive the complex factorization of the sequence {vn}with initial values v0 = 0 and v1 = 1 for r = 2.In Section 3, we solve the open problems in [3] for the sequence {vn} withinitial values v0 = 0 and v1 = 1 by using Chebyshev polynomials. Also, sincewe will get the complex factorization for any r, our results are a generalizationof [5].In Section 4, we provide new results by obtaining the binomial sum for thesequence {vn} by using Chebyshev polynomials of the second kind {Un(x)}.1. Chebyshev polynomials {Tn(x)} and {Un(x)}Chebyshev polynomials of the first and second kinds are the polynomials Tn(x)and Un(x), respectively, such thatTn(x) = cos(ncos−1 x)andUn(x) =sin((n+1)cos−1 x)sin(cos−1 x).Note that both formulas hold for all x where they make sense and they are de-fined by continuity for other values of x (since both formulas define polynomialsin the variable x at least on the interval−1 < x < 1). Also, Tn(x) and Un(x) bothsatisfy the following second order recurrenceyn+1(x) = 2xyn(x)− yn−1(x), n≥ 0,with initial conditionas T−1(x)= x, T0(x)= 1, T1(x)= x and U−1(x)= 0, U0(x)=1, U1(x) = 2x. The complex factorization of Chebyshev polynomials is given asfollows:Tn(x) = 2n−1n∏k=1(x− cos((2k−1)pi2n)), n≥ 1 (2)COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 181Un(x) = 2nn∏k=1(x− cos(kpin+1))(3)and it is well known that the binomial sums for Chebyshev polynomials areTn(x) =n2bn/2c∑j=0(−1) jn− j(n− jj)(2x)n−2 j. (4)andUn(x) =bn/2c∑j=0(−1) j(n− jj)xn−2 j. (5)Also, a well known relation between Chebsyshev polynomials isTn(x) =Un(x)− xUn−1(x). (6)(See [1], [2], [6], [9], [10] and [11] for details).2. The Connection between {vn} and {Un(x)}In this section, we give a connection between the sequence {vn} and Chebyshevpolynomials of second kind {Un(x)}. We need to remind that some definitionsfrom [8] to obtain our results.Consider the determinant of a tridiagonal matrix∆µ,ξ :=∣∣∣∣∣∣∣∣∣∣∣aµ 1−bµ+1 aµ+1 1. . . . . . . . .−bξ−1 aξ−1 1−bξ aξ∣∣∣∣∣∣∣∣∣∣∣if 0≤ µ < ξ ≤ rand if µ ≥ ξ ,∆µ,ξ :=0, if µ > ξ +11, if µ = ξ +1aµ , if µ = ξ .Then, we consider∆r :=∣∣∣∣∣∣∣∣∣∣∣∣∣a2 1 1−b3 a3 1. . . . . . . . .−br−1 ar−1 1−b0 a0 1−b2 −b1 a1∣∣∣∣∣∣∣∣∣∣∣∣∣.182 MURAT SAHIN - ELIF TAN - SEMIH YILMAZAlso, recall the following definitions from [8]:b := (−1)rr−1∏i=0bi,c := (−1)r (b2+b/b2)andU˜n(x) := dnUn(x− c2d), n≥ 0,where d is one of the square roots of b. (k in [8] corresponds to our r).Now, we can establish the connection between the sequence {vn} and Cheby-shev polynomials of the second kind {Un}.Lemma 2.1. For r ≥ 3, the terms of the sequence {vn} are given by in terms ofChebyshev polynomials of the second kind {Un} as follow:vnr = ∆2,rU˜n−1 (∆r) ,and for 1≤ t ≤ r−1,vnr+t = ∆2,tU˜n (∆r)+(−1)t(t+1∏i=2bi)∆t+2,rU˜n−1 (∆r) .Proof. We will use the results from [8] in the proof. Let {Rn+1(x)} be thesequence of polynomials defined by the recurrence relationRn+1(x) = (x−βn)Rn(x)− γnRn−1(x), n≥ 0,with initial conditions R−1(x) = 0 and R0(x) = 1 whereβnr+ j :=−a j+2,γnr+ j :=−b j+2, 0≤ j ≤ r−1, n≥ 0.Then clearly,vn = Rn−1(0), n≥ 0. (7)Let ∆µ,ξ (x) be a polynomial of degree ξ − µ + 1 obtained by replacing aiby x+ai in the definition of ∆µ,ξ . Similarly, let ϕr(x) be a polynomial of degreer obtained by replacing ai by x+ai in the definition of ∆r. In this case, ∆µ,ξ =∆µ,ξ (0) and ∆r = ϕr(0).We can obtainRnr+ j(x) = ∆2, j+1(x)U˜n(ϕr(x))+(−1) j+1(j+2∏i=2bi)∆ j+3,r(x)U˜n−1(ϕr(x)), (8)COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 183where 0≤ j ≤ r−1,n≥ 0, by using Theorem 5.1 in [4].If we use (7) and (8) then we getvnr = Rnr−1(0)= R(n−1)r+(r−1)(0) (Take j = r−1 and n = n−1 in (8))= ∆2,rU˜n−1(ϕr(0))+(−1)r(r+1∏i=2bi)∆r+2,r(0)U˜n−2(ϕr(0))= ∆2,rU˜n−1(∆r)+(−1)r(r+1∏i=2bi)∆r+2,rU˜n−2(∆r).Then, since ∆r+2,r = 0, we get the first equality in the hypothesis of theorem asfollow:vnr = ∆2,rU˜n−1(∆r).Now, again if we use (7) and (8) for 1≤ t ≤ r−1, we get the desired resultvnr+t = Rnr+t−1(0) (Take j = t−1 in (8))= ∆2,tU˜n(ϕr(0))+(−1)t(t+1∏i=2bi)∆t+2,r(0)U˜n−1(ϕr(0))= ∆2,tU˜n(∆r)+(−1)t(t+1∏i=2bi)∆t+2,rU˜n−1(∆r).Example 2.2. We combine Fibonacci, Jacobsthal and a second order recurrenceequations to get the following sequence {vn}:vn =vn−1+ vn−2, if n≡ 0(mod 3) ,vn−1+2vn−2, if n≡ 1(mod 3) ,3vn−1−2vn−2, if n≡ 2(mod 3) .A few terms of the sequence {vn} are {0,1,3,4,10,22,32,76,164,240,568,1224, . . .} and we have r = 3,a0 = a1 = b0 = 1, b1 = 2, a2 = 3 and b2 = −2.We need to compute ∆3, ∆2,1, ∆2,2, ∆2,3, b, c and d to establish the connection.By using the definitions from Section 2, we get∆3 =∣∣∣∣∣∣a2 1 1−b0 a0 1−b2 −b1 a1∣∣∣∣∣∣= 12, ∆2,3 =∣∣∣∣ a2 1−b0 a0∣∣∣∣= 4, ∆2,1 = 1, ∆2,2 = a2 = 3,184 MURAT SAHIN - ELIF TAN - SEMIH YILMAZTable 1:n v3n 2n+1Un−1(x)1 4 42 32 16x3 240 64x2−164 1792 256x3−128x5 13376 1024x4−768x2+646 99840 4096x5−4096x3+768xb = (−1)3b0b1b2 = 4, d =√b = 2, c = (−1)3(b2+bb2)= 4.Now, substituting these values in Lemma 2.1, we obtainv3n = ∆2,3U˜n−1(∆3)= ∆2,3dn−1Un−1(∆3− c2d)= 222n−1Un−1(12−44)= 2n+1Un−1 (2) .We show this connection in Table 1 by calculating a few terms of the se-quence of {vn} and Chebsyev polynomials of second kind {Un(x)}. We use asymbolic programming language to calculate the terms in the table.Similarly, we can obtain the connections for {v3n+1} and {v3n+2} by usingLemma 2.1.3. The Complex Factorization of the sequence {vn}Theorem 3.1. For r ≥ 3,vnr = ∆2,r(2d)n−1n−1∏k=1(∆r− c2d− cos(kpin)).Proof. If we use Lemma 2.1 and U˜n(x) := dnUn(x− c2d), we obtainvnr = ∆2,rU˜n−1(∆r)= ∆2,rdn−1Un−1(∆r− c2d).COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 185Now, if we use (3), that is the complex factorization of Chebyshev polyno-mials of the second kind {Un}, we get the desired resultvnr = ∆2,r(2d)n−1n−1∏k=1(∆r− c2d− cos(kpin)).Example 2.2 (continued). We get the connection between {v3n} and {Un}.So, using Theorem 3.1, we obtain the complex factorization of {v3n} as follow:v3n = ∆2,3(2d)n−1n−1∏k=1(∆3− c2d− cos(kpin))= 4n−1n−1∏k=1(12−44− cos(kpin))= 22n−2n−1∏k=1(2− cos(kpin)).Theorem 3.2. For r ≥ 3, if the equality∆2,t (c−∆r) = 2(−1)tt+1∏i=2bi∆t+2,r, 1≤ t ≤ r−1holds for some t theni.vnr+t = ∆2,tdnTn(∆r− c2d)ii..vnr+t = 2n−1dn∆2,tn∏k=1(∆r− c2d− cos((2k−1)pi2n)).Proof. We can obtain the following connection by using Lemma 2.1:vnr+t = ∆2,tdnUn(∆r− c2d)+(−1)tt+1∏i=2bi∆t+2,rdn−1Un−1(∆r− c2d)= dn∆2,t(Un(∆r− c2d)+(−1)t ∏t+1i=2 bi∆t+2,rd∆2,tUn−1(∆r− c2d)).186 MURAT SAHIN - ELIF TAN - SEMIH YILMAZAlso, if we substitute the equality∆2,t (c−∆r) = 2(−1)tt+1∏i=2bi∆t+2,r 1≤ t ≤ r−1,on the statement of theorem in the above equation, we obtainvnr+t = dn∆2,t(Un(∆r− c2d)− ∆r− c2dUn−1(∆r− c2d)).Now, if we use the well known identity Tn(x) = Un(x)− xUn−1(x) in the lastequation, we get the part (i) of the theorem:vnr+t = dn∆2,tTn(∆r− c2d).Now, by using Equation (2), that is the complex factorization of Chebyshevpolynomials of first kind we get the part (ii) of the theorem as follow:vnr+t = dn∆2,tTn(∆r− c2d)= 2n−1dn∆2,tn∏k=1(∆r− c2d− cos((2k−1)pi2n)).Example 3.3. Let us consider the following 3-periodic sequence {vn}vn =−7vn−1+6vn−2, if n≡ 0(mod 3) ,3vn−1−2vn−2, if n≡ 1(mod 3) ,vn−1+ vn−2, if n≡ 2(mod 3) .A few terms of the sequence {vn} are {0,1,1,−1,−5,−6,12,48,60,−132,−516,− 648,1440 . . .} and we have r = 3,a0 = −7,a1 = 3,a2 = 1 = b2,b0 = 6 andb1 = −2. We want to get the complex factorization of {v3n+2} by using Theo-rem 3.2. By using the definitions from Section 2, we get∆3 =∣∣∣∣∣∣a2 1 1−b0 a0 1−b2 −b1 a1∣∣∣∣∣∣=−25, ∆2,2 = ∆4,3 = 1,b = (−1)3b0b1b2 = 12, d =√b = 2√3, c = (−1)3(b2+bb2)=−13.COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 187For t = 2, since∆2,t (c−∆r)−2(−1)tt+1∏i=2bi∆t+2,r, = ∆2,2 (c−∆3)−2(−1)23∏i=2bi∆4,3= 1.(−13+25)−2.1.b2b3.1)= 12−2.b2b0= 0,the condition of Theorem 3.2 is satisfied. So, if we use Theorem 3.2, we obtainv3n+2 = ∆2,tdnTn(∆r− c2d)= ∆2,2(2√3)nTn(∆3− (−13)4√3)= (2√3)nTn(−√3).and we get the complex factorizationv3n+2 = 2n−1dn∆2,tn∏k=1(∆r− c2d− cos((2k−1)pi2n))= ∆2,22n−1dnn∏k=1(∆3− c2d− cos((2k−1)pi2n))= ∆2,22n−1(2√2)nn∏k=1(−25+134√2− cos((2k−1)pi2n))= 2(5n−2)/2n∏k=1(−3√2− cos((2k−1)pi2n)).4. The Binomial Sum for the sequence {vn}Theorem 4.1. For r > 3, {vnr} can be defined in terms of sumsvnr = ∆2,rdn−1[(n−1)/2]∑j=0(−1) j(n−1− jj)(∆r− c2d)n−1−2 j.Proof. We have the connectionvnr = ∆2,rdn−1Un−1(∆r− c2d)188 MURAT SAHIN - ELIF TAN - SEMIH YILMAZby Lemma 2.1. If we make a substitution using (5) in this connection, we obtainthe desired resultvnr = ∆2,rdn−1[(n−1)/2]∑j=0(−1) j(n−1− jj)(∆r− c2d)n−1−2 j.We can get the binomial sum for the sequence {v3n} in the Example 2.2 asan example:Example 2.2 (continued). Bu using Theorem 4.1, we can write the sequence{v3n} in terms of sums as follow:vnr = ∆2,rdn−1[(n−1)/2]∑j=0(−1) j(n−1− jj)(∆r− c2d)n−1−2 j= ∆2,32n−1[(n−1)/2]∑j=0(−1) j(n−1− jj)(∆3−44)n−1−2 j= 2n+1[(n−1)/2]∑j=0(−1) j(n−1− jj)2n−1−2 j.REFERENCES[1] D. Aharonov, A. Beardon and K. Driver, Fibonacci, Chebyshev, and OrthogonalPolynomials, The American Mathematical Monthly, Vol. 112 (2005), No. 7, 612–630.[2] N. Cahill, D. Derrico and J. R. Spence, J. P., Complex factorizations of the Fi-bonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13-19.[3] C. Cooper and R. Parry Jr., Factorizations of some periodic linear recurrence sys-tems, The Eleventh International Conference on Fibonacci Numbers and TheirApplications, Germany, July 2004.[4] M. N. de Jesus and J. Petronilho, On orthogonal polynomials obtained via poly-nomial mappings, J. Approx. Theory 162 (2010), 2243-2277.[5] Song Pyo Jun, Complex factorization of the generalized fibonacci sequences {qn},Korean J. Math. 23 (2015), no. 3, 371–377.COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 189[6] Russell Hendel and Charlie Cook, Recursive properties of trigonemetric products,App. of Fib. Numbers (1996), 6, 201–214.[7] D. Panario, M. Sahin and Q. Wang, A family of Fibonacci-like conditional se-quences, INTEGERS Electronic Journal of Combinatorial Number Theory, Sec-ond Revision, 2012.[8] J. Petronilho, Generalized Fibonacci sequences via ortogonal polynomials, Ap-plied Mathematics and Computation, 218 (2012), 9819–9824.[9] T.J. Rivlin, The Chebyshev Polynomials-From Approximation Theory to Algebraand Number Theory, Wiley-Interscience, John Wiley, 1990.[10] E. Weisstein, Chebyshev Polynomial of the FirstKind, From MathWorld–A Wolfram Web Resource.http://mathworld.wolfram.com/ChebyshevPolynomialoftheFirstKind.html[11] E. Weisstein, Chebyshev Polynomial of the Sec-ond Kind, From MathWorld–A Wolfram Web Resource,http://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.htmlMURAT SAHINDepartment of Mathematics, Science FacultyAnkara Universitye-mail: msahin@ankara.edu.trELIF TANDepartment of Mathematics, Science FacultyAnkara Universitye-mail: etan@ankara.edu.trSEMIH YILMAZDepartment of Actuarial ScienceKirikkale Universitye-mail: syilmaz@kku.edu.tr

Complex Factorization by Chebysev Polynomials

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Complex Factorization by Chebysev Polynomials

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Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)