On Generalized Fibonacci Numbers

Fidel Oduol; Isaac Owino Okoth

doi:10.33434/cams.771023

EN

On Generalized Fibonacci Numbers

Abstract

Fibonacci numbers and their polynomials have been generalized mainly by two ways: by maintaining the recurrence relation and varying the initial conditions, and by varying the recurrence relation and maintaining the initial conditions. In this paper, we introduce and derive various properties of $r$-sum Fibonacci numbers. The recurrence relation is maintained but initial conditions are varied. Among results obtained are Binet's formula, generating function, explicit sum formula, sum of first $n$ terms, sum of first $n$ terms with even indices, sum of first $n$ terms with odd indices, alternating sum of $n$ terms of $r-$sum Fibonacci sequence, Honsberger's identity, determinant identities and a generalized identity from which Cassini's identity, Catalan's identity and d'Ocagne's identity follow immediately.

Keywords

Binet's formula , Fibonacci sequence , generating function , r-shifted Fibonacci sequence

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Fidel Oduol This is me
0000-0002-1228-6339
Kenya

Isaac Owino Okoth ^*
Kenya

Publication Date

December 22, 2020

Submission Date

July 17, 2020

Acceptance Date

September 29, 2020

Published in Issue

Year 2020 Volume: 3 Number: 4

DOI

https://doi.org/10.33434/cams.771023

IZ

https://izlik.org/JA86FK54HE

APA

Oduol, F., & Okoth, I. O. (2020). On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences, 3(4), 186-202. https://doi.org/10.33434/cams.771023

AMA

1.Oduol F, Okoth IO. On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences. 2020;3(4):186-202. doi:10.33434/cams.771023

Chicago

Oduol, Fidel, and Isaac Owino Okoth. 2020. “On Generalized Fibonacci Numbers”. Communications in Advanced Mathematical Sciences 3 (4): 186-202. https://doi.org/10.33434/cams.771023.

EndNote

Oduol F, Okoth IO (December 1, 2020) On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences 3 4 186–202.

IEEE

[1]F. Oduol and I. O. Okoth, “On Generalized Fibonacci Numbers”, Communications in Advanced Mathematical Sciences, vol. 3, no. 4, pp. 186–202, Dec. 2020, doi: 10.33434/cams.771023.

ISNAD

Oduol, Fidel - Okoth, Isaac Owino. “On Generalized Fibonacci Numbers”. Communications in Advanced Mathematical Sciences 3/4 (December 1, 2020): 186-202. https://doi.org/10.33434/cams.771023.

JAMA

1.Oduol F, Okoth IO. On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences. 2020;3:186–202.

MLA

Oduol, Fidel, and Isaac Owino Okoth. “On Generalized Fibonacci Numbers”. Communications in Advanced Mathematical Sciences, vol. 3, no. 4, Dec. 2020, pp. 186-02, doi:10.33434/cams.771023.

Vancouver

1.Fidel Oduol, Isaac Owino Okoth. On Generalized Fibonacci Numbers. Communications in Advanced Mathematical Sciences. 2020 Dec. 1;3(4):186-202. doi:10.33434/cams.771023