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Mathematica Slovaca

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Volume 68, Issue 4


A multi-parameter generalization of the symmetric algorithm

José L. Ramírez / Mark Shattuck
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  • Institute for Computational Science & Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
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Published Online: 2018-08-06 | DOI: https://doi.org/10.1515/ms-2017-0137


The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

MSC 2010: Primary 05A15; Secondary 05A19

Keywords: symmetric algorithm; Euler-Seidel method; q-analogue; q-Fibonacci numbers; combinatorial identities


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About the article

Received: 2016-07-01

Accepted: 2017-02-12

Published Online: 2018-08-06

Published in Print: 2018-08-28

Communicated by Stanislav Jakubec

Citation Information: Mathematica Slovaca, Volume 68, Issue 4, Pages 699–712, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0137.

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