COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS

Transcription

1 LE MATEMATICHE Vol LXXIII 2018 Fasc I, pp doi: / COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS MURAT SAHIN - ELIF TAN - SEMIH YILMAZ Let {a i },{b i } be real numbers for 0 i r 1, and define a r- periodic sequence {v n } with initial conditions v 0, v 1 and recurrences v n = a t v + b t v n 2 where n t mod r n 2 In this paper, by aid of Chebyshev polynomials, we introduce a new method to obtain the complex factorization of the sequence {v n } so that we extend some recent results and solve some open problems Also, we provide new results by obtaining the binomial sum for the sequence {v n } by using Chebyshev polynomials Let {a i } and {b i } be real numbers for 0 i r 1, and define a sequence {v n } with initial conditions v 0, v 1, and for n 2, a 0 v + b 0 v n 2, a 1 v + b 1 v n 2, v n = a r 1 v + b r 1 v n 2, if n 0mod r, if n 1mod r, if n r 1mod r We call {v n } as a r-periodic sequences It is studied in [7] by Panario et al and they find the generating function and Binet s like formula for the sequence {v n } via generalized continuant Petronilho obtain the same Binet s like formula by using tools from ortogonal polynomials in [8] Entrato in redazione: 1 gennaio 2007 AMS 2010 Subject Classification: 33C47,65Q30,11B39,05A15 Keywords: Chebyshev polynomials, complex factorization, binomial sums 1

2 180 MURAT SAHIN - ELIF TAN - SEMIH YILMAZ For r = 2 and initial values v 0 = 1, v 1 = a 1, Cooper and Parry [3] called the sequence {v n } as the period two second order linear recurrence system, and gave the complex factorization of odd terms of this sequence by determining the eigenvalues and eigenvectors of certain tridiagonal matrices The problems remained unsolved in [3] are determining the complex factorization of even terms of the period two second order linear recurrence system and determining the complex factorization of the sequence {v n } for given general r Also, for r = 2 and initial values v 0 = 0, v 1 = 1, Jun [5] give a connection between the sequence {v n } and Chebyshev polynomials of the second kind {U n x} By using the factorization of {U n x}, Jun derive the complex factorization of the sequence {v n } with initial values v 0 = 0 and v 1 = 1 for r = 2 In Section 3, we solve the open problems in [3] for the sequence {v n } with initial values v 0 = 0 and v 1 = 1 by using Chebyshev polynomials Also, since we will get the complex factorization for any r, our results are a generalization of [5] In Section 4, we provide new results by obtaining the binomial sum for the sequence {v n } by using Chebyshev polynomials of the second kind {U n x} 1 Chebyshev polynomials {T n x} and {U n x} Chebyshev polynomials of the first and second kinds are the polynomials T n x and U n x, respectively, such that and T n x = cos ncos 1 x U n x = sinn + 1cos 1 x sincos 1 x Note that both formulas hold for all x where they make sense and they are defined by continuity for other values of x since both formulas define polynomials in the variable x at least on the interval 1 < x < 1 Also, T n x and U n x both satisfy the following second order recurrence y n+1 x = 2xy n x y x, n 0, with initial conditionas T 1 x = x, T 0 x = 1, T 1 x = x and U 1 x = 0, U 0 x = 1, U 1 x = 2x The complex factorization of Chebyshev polynomials is given as follows: T n x = 2 n x cos 2k 1π, n 1 2

3 COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 181 U n x = 2 n n kπ x cos n + 1 and it is well known that the binomial sums for Chebyshev polynomials are T n x = n 1 j n j 2x n 2 j 4 2 n j j and n/2 n/2 U n x = 1 j n j j 3 x n 2 j 5 Also, a well known relation between Chebsyshev polynomials is See [1], [2], [6], [9], [10] and [11] for details T n x = U n x xu x 6 2 The Connection between {v n } and {U n x} In this section, we give a connection between the sequence {v n } and Chebyshev polynomials of second kind {U n x} We need to remind that some definitions from [8] to obtain our results Consider the determinant of a tridiagonal matrix µ,ξ := and if µ ξ, Then, we consider r := a µ 1 b µ+1 a µ+1 1 µ,ξ := b ξ 1 a ξ 1 1 b ξ a ξ 0, if µ > ξ + 1 1, if µ = ξ + 1 a µ, if µ = ξ a b 3 a 3 1 b r 1 a r 1 1 b 0 a 0 1 b 2 b 1 a 1 if 0 µ < ξ r

4 182 MURAT SAHIN - ELIF TAN - SEMIH YILMAZ Also, recall the following definitions from [8]: and b := 1 r r 1 b i, i=0 c := 1 r b 2 + b/b 2 x c Ũ n x := d n U n, 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 {v n } and Chebyshev polynomials of the second kind {U n } Lemma 21 For r 3, the terms of the sequence {v n } are given by in terms of Chebyshev polynomials of the second kind {U n } as follow: and for 1 t r 1, v nr = 2,r Ũ r, t+1 v nr+t = 2,t Ũ n r + 1 t b i t+2,r Ũ r Proof We will use the results from [8] in the proof Let {R n+1 x} be the sequence of polynomials defined by the recurrence relation R n+1 x = x β n R n x γ n R x, n 0, with initial conditions R 1 x = 0 and R 0 x = 1 where β nr+ j := a j+2,γ nr+ j := b j+2, 0 j r 1, n 0 Then clearly, v n = R 0, n 0 7 Let µ,ξ x be a polynomial of degree ξ µ + 1 obtained by replacing a i by x+a i in the definition of µ,ξ Similarly, let ϕ r x be a polynomial of degree r obtained by replacing a i by x + a i in the definition of r In this case, µ,ξ = µ,ξ 0 and r = ϕ r 0 We can obtain j+2 R nr+ j x = 2, j+1 xũ n ϕ r x+ 1 j+1 b i j+3,r xũ ϕ r x, 8

5 COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 183 where 0 j r 1,n 0, by using Theorem 51 in [4] If we use 7 and 8 then we get v nr = R nr 1 0 = R r+r 1 0 Take j = r 1 and n = n 1 in 8 r+1 = 2,r Ũ ϕ r r b i r+2,r 0Ũ n 2 ϕ r 0 r+1 = 2,r Ũ r + 1 r b i r+2,r Ũ n 2 r Then, since r+2,r = 0, we get the first equality in the hypothesis of theorem as follow: v nr = 2,r Ũ r Now, again if we use 7 and 8 for 1 t r 1, we get the desired result v nr+t = R nr+t 1 0 Take j = t 1 in 8 t+1 = 2,t Ũ n ϕ r t b i t+2,r 0Ũ ϕ r 0 t+1 = 2,t Ũ n r + 1 t b i t+2,r Ũ r Example 22 We combine Fibonacci, Jacobsthal and a second order recurrence equations to get the following sequence {v n }: v + v n 2, if n 0mod 3, v n = v + 2v n 2, if n 1mod 3, 3v 2v n 2, if n 2mod 3 A few terms of the sequence {v n } are {0,1,3,4,10,22,32,76,164,240,568, 1224,} and we have r = 3,a 0 = a 1 = b 0 = 1, b 1 = 2, a 2 = 3 and b 2 = 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 = a b 0 a 0 1 b 2 b 1 a 1 = 12, 2,3 = a 2 1 b 0 a 0 = 4, 2,1 = 1, 2,2 = a 2 = 3,

6 184 MURAT SAHIN - ELIF TAN - SEMIH YILMAZ Table 1: n v 3n 2 n+1 U x x x x 3 128x x 4 768x x x x b = 1 3 b 0 b 1 b 2 = 4, d = b = 2, c = 1 3 b 2 + b b 2 = 4 Now, substituting these values in Lemma 21, we obtain v 3n = 2,3 Ũ 3 = 2,3 d 3 c U 12 4 = U 4 = 2 n+1 U 2 We show this connection in Table 1 by calculating a few terms of the sequence of {v n } and Chebsyev polynomials of second kind {U n x} We use a symbolic programming language to calculate the terms in the table Similarly, we can obtain the connections for {v 3n+1 } and {v 3n+2 } by using Lemma 21 3 The Complex Factorization of the sequence {v n } Theorem 31 For r 3, v nr = 2,r r c kπ cos n Proof If we use Lemma 21 and Ũ n x := d n U n x c v nr = 2,r Ũ r = 2,r d U r c, we obtain

7 COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 185 Now, if we use 3, that is the complex factorization of Chebyshev polynomials of the second kind {U n }, we get the desired result v nr = 2,r r c cos kπ n Example 22 continued We get the connection between {v 3n } and {U n } So, using Theorem 31, we obtain the complex factorization of {v 3n } as follow: 3 c kπ v 3n = 2,3 cos n 12 4 kπ = 4 cos 4 n kπ = 2 2 cos n 2 Theorem 32 For r 3, if the equality holds for some t then 2,t c r = 2 1 t t+1 b i t+2,r, 1 t r 1 iiv nr+t = 2 d n 2,t n iv nr+t = 2,t d n r c T n r c 2k 1π cos Proof We can obtain the following connection by using Lemma 21: v nr+t = 2,t d n r c U n + 1 t t+1 b i t+2,r d r c U = d n r c 2,t U n + 1t t+1 b i t+2,r r c U d 2,t

8 186 MURAT SAHIN - ELIF TAN - SEMIH YILMAZ Also, if we substitute the equality 2,t c r = 2 1 t t+1 b i t+2,r 1 t r 1, on the statement of theorem in the above equation, we obtain v nr+t = d n r c 2,t U n r c U r c Now, if we use the well known identity T n x = U n x xu x in the last equation, we get the part i of the theorem: v nr+t = d n r c 2,t T n Now, by using Equation 2, that is the complex factorization of Chebyshev polynomials of first kind we get the part ii of the theorem as follow: v nr+t = d n r c 2,t T n = 2 d n n r c 2k 1π 2,t cos Example 33 Let us consider the following 3-periodic sequence {v n } 7v + 6v n 2, if n 0mod 3, v n = 3v 2v n 2, if n 1mod 3, v + v n 2, if n 2mod 3 A few terms of the sequence {v n } are {0,1,1, 1, 5, 6,12,48,60, 132, 516, 648,1440} and we have r = 3,a 0 = 7,a 1 = 3,a 2 = 1 = b 2,b 0 = 6 and b 1 = 2 We want to get the complex factorization of {v 3n+2 } by using Theorem 32 By using the definitions from Section 2, we get 3 = a b 0 a 0 1 b 2 b 1 a 1 = 25, 2,2 = 4,3 = 1, b = 1 3 b 0 b 1 b 2 = 12, d = b = 2 3, c = 1 3 b 2 + b b 2 = 13

9 COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 187 For t = 2, since 2,t c r 2 1 t t+1 b i t+2,r, = 2,2 c b i 4,3 = b 2 b 3 1 = 12 2b 2 b 0 = 0, the condition of Theorem 32 is satisfied So, if we use Theorem 32, we obtain v 3n+2 = 2,t d n r c T n and we get the complex factorization = 2,2 2 3 n T n = 2 3 n T n 3 v 3n+2 = 2 d n n r c 2k 1π 2,t cos = 2,2 2 d n n 3 c 2k 1π cos = 2, n n k 1π 4 cos 2 = 2 5n 2/2 n 3 2k 1π cos 2 4 The Binomial Sum for the sequence {v n } Theorem 41 For r > 3, {v nr } can be defined in terms of sums [/2] n 1 j v nr = 2,r d 1 j r c 2 j j Proof We have the connection v nr = 2,r d r c U

10 188 MURAT SAHIN - ELIF TAN - SEMIH YILMAZ by Lemma 21 If we make a substitution using 5 in this connection, we obtain the desired result [/2] n 1 j v nr = 2,r d 1 j r c 2 j j We can get the binomial sum for the sequence {v 3n } in the Example 22 as an example: Example 22 continued Bu using Theorem 41, we can write the sequence {v 3n } in terms of sums as follow: [/2] n 1 j v nr = 2,r d 1 j r c j [/2] n 1 j = 2,3 2 1 j 3 4 j 4 [/2] n 1 j = 2 n+1 1 j 2 2 j j 2 j 2 j REFERENCES [1] D Aharonov, A Beardon and K Driver, Fibonacci, Chebyshev, and Orthogonal Polynomials, The American Mathematical Monthly, Vol , No 7, [2] N Cahill, D Derrico and J R Spence, J P, Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart , [3] C Cooper and R Parry Jr, Factorizations of some periodic linear recurrence systems, The Eleventh International Conference on Fibonacci Numbers and Their Applications, Germany, July 2004 [4] M N de Jesus and J Petronilho, On orthogonal polynomials obtained via polynomial mappings, J Approx Theory , [5] Song Pyo Jun, Complex factorization of the generalized fibonacci sequences {q n }, Korean J Math , no 3,

11 COMPLEX FACTORIZATION BY CHEBYSEV POLYNOMIALS 189 [6] Russell Hendel and Charlie Cook, Recursive properties of trigonemetric products, App of Fib Numbers 1996, 6, [7] D Panario, M Sahin and Q Wang, A family of Fibonacci-like conditional sequences, INTEGERS Electronic Journal of Combinatorial Number Theory, Second Revision, 2012 [8] J Petronilho, Generalized Fibonacci sequences via ortogonal polynomials, Applied Mathematics and Computation, , [9] TJ Rivlin, The Chebyshev Polynomials-From Approximation Theory to Algebra and Number Theory, Wiley-Interscience, John Wiley, 1990 [10] E Weisstein, Chebyshev Polynomial of the First Kind, From MathWorld A Wolfram Web Resource [11] E Weisstein, Chebyshev Polynomial of the Second Kind, From MathWorld A Wolfram Web Resource, MURAT SAHIN Department of Mathematics, Science Faculty Ankara University msahin@ankaraedutr ELIF TAN Department of Mathematics, Science Faculty Ankara University etan@ankaraedutr SEMIH YILMAZ Department of Actuarial Science Kirikkale University syilmaz@kkuedutr