Trigonometric Factorizations of the Pell and Jacobsthal Sequences

Abstract

The Pell and Jacobsthal numbers have inspired many studies in mathematics, especially in number theory. In this paper, we derive the trigonometric factorizations of the Pell and Jacobsthal sequences by using determinants of two specific new tridiagonal matrices and the roots of the Chebyshev polynomial of the second kind. Furthermore, we provide two Maple procedures to calculate the trigonometric factorizations of these integer sequences.

Keywords

• Pell sequences
• Jacobsthal sequences
• factorizations
• tridiagonal matrices
• determinants

References

1. T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley & Sons, 2001.
2. T. Koshy, Fibonacci, Lucas, and Pell numbers, and Pascal's triangle, Mathematical Spectrum 43 (3) (2011) 125–132.
3. N. J. A. Sloane, The on-line encyclopedia of integer sequences (1964), http://oeis.org, Accessed 01 Feb 2024.
4. N. D. Cahill, J. R. D'Ericco, J. P. Spence, Complex factorizations of the Fibonacci and Lucas numbers, The Fibonacci Quarterly 41 (1) (2003) 13–19.
5. S. B. Bozkurt, F. Yılmaz, D. Bozkurt, On the complex factorization of the Lucas sequence, Applied Mathematics Letters 24 (8) (2011) 1317–1321.
6. A. Öteleş, M. Akbulak, Positive integer powers of one type of complex tridiagonal matrices, Bulletin of the Malaysian Mathematical Sciences Society 37 (4) (2014) 971–988.
7. A. Öteleş, M. Akbulak, Positive integer powers of certain complex tridiagonal matrices, Applied Mathematics and Computation 219 (21) (2013) 10448–10455.
8. H. Wu, Complex factorizations of the Lucas sequences via matrix methods, Journal of Applied Mathematics 2014 (2014) 387675.

9. E. Kilic, D. Tasci, Negatively subscripted Fibonacci and Lucas numbers and their complex factorizations, Ars Combinatoria 96 (2010) 275–288.
10. E. Kilic, P. Stanica, Factorizations and representations of second linear recurrences with indices in arithmetic progressions, Bulletin of the Mexican Mathematical Society 15 (1) (2009) 23–26.
11. J. Seibert, P. Trojovsky, On factorization of the Fibonacci and Lucas numbers using tridiagonal determinants, Mathematica Slovaca 62 (3) (2012) 439–450.
12. J. Seibert, P. Trojovsky, On factorization of the generalized Fibonacci numbers, International Journal of Pure and Applied Mathematics 30 (1) (2006) 23–32.
13. C. M. da Fonseca, J. Petronilho, Explicit inverses of some tridiagonal matrices, Linear Algebra and its Applications 325 (1-3) (2001) 7–21.
14. L. Fox, J. B. Parke, Chebyshev polynomials in numerical analysis, Oxford University Press, 1968.