A General Convolution Identity

A general convolution identity is derived for Fibonaccitype and Lucastype sequences. The convolution of the Fibonacci and Lucas numbers is shown to be equal to a simple multiple of a Fibonacci number. This result is then generalized to other sequences such as the Pell, Padovan, and Tribonacci number...

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Published in:Mathematics magazine 2024-04, Vol.97 (2), p.98
Main Authors: Dresden, Greg, Wang, Yichen
Format: Magazinearticle
Language:English
Subjects:
Convolution
Fibonacci numbers
Identities
Identity
Mathematics
Numbers
Sequences
Theorems
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Wang, Yichen
description A general convolution identity is derived for Fibonaccitype and Lucastype sequences. The convolution of the Fibonacci and Lucas numbers is shown to be equal to a simple multiple of a Fibonacci number. This result is then generalized to other sequences such as the Pell, Padovan, and Tribonacci numbers. The concept of generating functions is introduced and used to prove the convolution identity. Newton's identities are also discussed in the context of these sequences. The paper concludes by mentioning other convolution formulas that have been discovered and the potential for further research in this area.
doi_str_mv 10.1080/0025570X
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subjects Convolution
Fibonacci numbers
Identities
Identity
Mathematics
Numbers
Sequences
Theorems
title A General Convolution Identity
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