New Generalizations of Fibonacci and Lucas Polynomials

Article Sidebar

PDF
Published: Jun 28, 2022
Keywords:
Fibonacci polynomial Lucas polynomial Generating function Binet’s formula

Main Article Content

Poonchayar Patthanangkoor
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University
Chonticha Chinsaard
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12120
Wisinee Ngamsriwiset
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12120
Thofun Waewkrathok
Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani 12120

Abstract

We consider the polynomials fn(x)  and ln(x)   which are generated by the recurrence relations fn(x) = 2ax fn-1(x)+(b-a2) fn-2(x)   for n >=2  and ln(x) = 2ax ln-1(x)+(b-a2) ln-2(x)  for n >=2 with the initial conditions f0(x) =0 , f1(x) =1 and  l0(x) =2 , l1(x) =2ax  where  a and  b are any non-zero real numbers. We obtain the new generalizations of Fibonacci and Lucas polynomials. Moreover, we obtain generating functions, Binet’s formulas and some identities involving  fn(x)  and ln(x) .

Article Details

Issue
Vol.30 No.3 (May - June 2022)
Section
Physical Sciences

References

Bicknell, M., 1970, A primer for the Fibonacci numbers VII, Fibonacci Quarterly. 8: 407-420.

Bilgici, G., 2014, New Generalizations of Fibonacci and Lucas Sequence, Applied Mathematical Sciences. 8(29): 1429-1437.

Edson, M. and Yayenie, O., 2009, A New Generalization of Fibonacci Sequences and Extended Binet’s Formula, Integers. 9: 639-654.

Horadam, A. F. and Mahon, Bro. J. M., 1983, Pell and Pell-Lucas Polynomials, Univerity of New England Armidale. Catholic Collage of Education. Sydney. Australia.

Jacobsthal, E., 1957, Über eine zahlentheoretische Summe, Norske Vid. Selsk. Forh. Trondheim. 30: 35-41.