A NEW GENERALIZED FIBONACCI SEQUENCE AND ITS PROPERTIES
By
1Manjeet Singh Teeth, 2Garvita Agrawal and 3K. N. Rajeswari
1,3Department of Mathematics, M.B. Khalsa College, Indore-452001, Madhya Pradesh, India.
2School of Mathematics, Devi Ahilya Vishwavidyalaya, Indore-452001, Madhya Pradesh, India.
Email:India.manjeetsinghteeth@rediffmail.com, gravyagrawal@gmail.com, knr_k@yahoo.co.in
(Received : October 28,2020 ; In format : August 14,2021; Revised in final form : November 03, 2021)
DOI: https://doi.org/10.58250/Jnanabha.2021.51205
Abstract
The generalized Fibonacci sequence is defined as Fk = pFk-1 + qFk-2; k ≥2 with F0 = a; F1 = b where p, q, a, b all are positive integers. Panwar et al. derived the sequence {Vk}k≥0 which is generated by taking p = 1; q = a = b = 2 then Vk = Vk-1+ 2Vk-2; k ≥2 with V0= 2; V1 = 2 and {Uk}k≥0 which is generated by taking p = 1; q = a = 2; b = 0 then Uk = Uk-1+ 2Uk-2; k ≥2 with U0= 2; U1 = 0. The generalization of the Fibonacci sequence can be done in many ways by changing the initial condition and others by changing the recurrence relation. We introduce the more general sequence {Sk}k≥0 by taking p = 1, q = b = n, a = 0 then we have Sk = Sk-1+ nSk-2; k ≥2 with S0= 2; S1 = 0
In this paper, we proved properties like Catalan’s identity, Cassini’s identity, d’Ocagane’s identity, Vajda’s identity and A variant identity for our sequence.
2020 Mathematical Sciences Classification: 11B39.
Keywords and Phrases: Fibonacci sequence, Recursive Relation.