Accessible Requires Authentication Published by De Gruyter August 6, 2018

A multi-parameter generalization of the symmetric algorithm

José L. Ramírez and Mark Shattuck
From the journal Mathematica Slovaca

Abstract

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.

  1. Communicated by Stanislav Jakubec

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Received: 2016-07-01
Accepted: 2017-02-12
Published Online: 2018-08-06
Published in Print: 2018-08-28

© 2018 Mathematical Institute Slovak Academy of Sciences