Trigonometric Pseudo Fibonacci Sequence

C. N. Phadte and S. P. Pethe
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 21, 2015, Number 3, Pages 70–76
Full paper (PDF, 171 Kb)

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Authors and affiliations

C. N. Phadte
Department of Mathematics, Goa University
Taleigao Goa, India

S. P. Pethe
Dr. Flat No.1 Premsagar Society
Mahatmanagar, Road D-2, Nasik-442007, India

Abstract

In this paper, we establish some results about second order non homogeneous recurrence relation containing extended trignometric function. Earlier {4}, we proved some properties of recurrence relation
g_n+2 = g_n+1 + g_n + Atⁿ, n = 0, 1, … with g₀ = 0, g₁ = 1; where both A ≠ 0 and t ≠ 0, and also t ≠ α, β where α, β are the roots of x² − x − 1 = 0.
Using the properties of generalised circular functions and Elmore’s method, we define a new sequence {H_n} which is the extension of Pseudo Fibonacci Sequence, given by recurrence relation
H_n+2 = pH_n+1 − qH_n + RtⁿN_r,0(t^*x),
where N_r,0(t^*x) is extended circular function.
We state and prove some properties for this extended Pseudo Fibonacci Sequence {H_n}.

Keywords

• Pseudo Fibonacci Sequence
• Non-homogeneous recurrence relation

AMS Classification

• 11B39

References

1. Elmore, M. (1967) Fibonacci functions, Fibonacci Quarterly 4, 5, 371–382.
2. Mikusinski, J. G. (1948) Sur les Fonctions, Annales da la Societe Polonaize de Mathematique, 21, 46–51.
3. Horadam, A. F. (1965) Basic Property of a certain Generalized Sequence of Numbers, Fibonacci Quarterly, 3(3), 161–176.
4. Phadte, C. N., & Pethe S. P. (2013) On Second Order Non-Homogeneous Recurrence Relation, Annales Mathematicae et Informaticae 41, 205–210.
5. Pethe, S. P., & Phadte C. N. (1993) Generalization of the Fibonacci Sequence, Applications of Fibonacci Numbers, Kluwer Academic Pub., 5, 465–472.
6. Walton, J. E., & Horadam A. F. (1974) Some Aspect of Fibonacci Numbers. The Fibonacci Quarterly, 12(3), 241–250.