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 F_k = pF_k-1 + qF_k-2; k ≥2 with F₀ = a; F₁ = b where p, q, a, b all are positive integers. Panwar et al. derived the sequence {V_k}_k≥0 which is generated by taking p = 1; q = a = b = 2 then V_k = V_k-1+ 2V_k-2; k ≥2 with V₀= 2; V₁ = 2 and {U_k}_k≥0 which is generated by taking p = 1; q = a = 2; b = 0 then U_k = U_k-1+ 2U_k-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 {S_k}_k≥0 by taking p = 1, q = b = n, a = 0 then we have S_k = S_k-1+ nS_k-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.

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