Abstract:a b s t r a c tIn this paper, firstly we present a connection between determinants of tridiagonal matrices and the Lucas sequence. Secondly, we obtain the complex factorization of Lucas sequence by considering how the Lucas sequence can be connected to Chebyshev polynomials by determinants of a sequence of matrices.
“…), (55), (56), and (50), we derive ( , ) = 2(√ ) cos ( arccos ( For = 1, = −1, (38) and (39) are exactly the formulas for Lucas numbers in [2].Remark 7. For = −1, (38) and (39) are exactly the formulas for { ( , −1)} in[4].…”
mentioning
confidence: 87%
“…The complex factorization of { ( , −1)} ≥0 was derived in [4], and we generalize the formula to { ( , )} ≥0 for any integer in the following theorem.…”
Section: Complex Factorizations Of the Second Lucas Sequencementioning
confidence: 99%
“…by Theorem 1 of [4]. Unlike the derivation in Theorem 3, we will not compute the spectrum of ( ) directly.…”
Section: Complex Factorizations Of the Second Lucas Sequencementioning
confidence: 99%
“…There is a long tradition of using matrix methods to study Lucas sequences [2][3][4][5]. In 2003, Cahill et al obtained complex factorizations for Fibonacci numbers and Lucas numbers by using the determinants of two slightly different sequences of tridiagonal matrices [2].…”
Section: Introductionmentioning
confidence: 99%
“…where is the th Lucas number and i = √ −1. In 2011, Burcu Bozkurt et al obtained the complex factorization of the second Lucas sequence { ( , −1)} ≥0 [4]:…”
Firstly, we show a connection between the first Lucas sequence and the determinants of some tridiagonal matrices. Secondly, we derive the complex factorizations of the first Lucas sequence by computing those determinants with the help of Chebyshev polynomials of the second kind. Furthermore, we also obtain the complex factorizations of the second Lucas sequence by the similar matrix method using Chebyshev polynomials of the first kind.
“…), (55), (56), and (50), we derive ( , ) = 2(√ ) cos ( arccos ( For = 1, = −1, (38) and (39) are exactly the formulas for Lucas numbers in [2].Remark 7. For = −1, (38) and (39) are exactly the formulas for { ( , −1)} in[4].…”
mentioning
confidence: 87%
“…The complex factorization of { ( , −1)} ≥0 was derived in [4], and we generalize the formula to { ( , )} ≥0 for any integer in the following theorem.…”
Section: Complex Factorizations Of the Second Lucas Sequencementioning
confidence: 99%
“…by Theorem 1 of [4]. Unlike the derivation in Theorem 3, we will not compute the spectrum of ( ) directly.…”
Section: Complex Factorizations Of the Second Lucas Sequencementioning
confidence: 99%
“…There is a long tradition of using matrix methods to study Lucas sequences [2][3][4][5]. In 2003, Cahill et al obtained complex factorizations for Fibonacci numbers and Lucas numbers by using the determinants of two slightly different sequences of tridiagonal matrices [2].…”
Section: Introductionmentioning
confidence: 99%
“…where is the th Lucas number and i = √ −1. In 2011, Burcu Bozkurt et al obtained the complex factorization of the second Lucas sequence { ( , −1)} ≥0 [4]:…”
Firstly, we show a connection between the first Lucas sequence and the determinants of some tridiagonal matrices. Secondly, we derive the complex factorizations of the first Lucas sequence by computing those determinants with the help of Chebyshev polynomials of the second kind. Furthermore, we also obtain the complex factorizations of the second Lucas sequence by the similar matrix method using Chebyshev polynomials of the first kind.
Abstract. We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type 2 F 1 . Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivatives sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials.
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