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A Multi-Parameter Generalization of the Symmetric Algorithm

Authors: 

Jose L. Ramırez, Mark Shattuck*

Source title: 
Mathematica Slovaca, 68(4): 699-712, 2018 (ISI)
Academic year of acceptance: 
2018-2019
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.