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A multi-parameter generalization of the symmetric algorithm

In: Mathematica Slovaca, vol. 68, no. 4
José Ramírez - Mark Shattuck
Detaily:
Rok, strany: 2018, 699 - 712
Kľúčové slová:
symmetric algorithm, Euler-Seidel method, $q$-analogue, $q$-Fibonacci numbers, combinatorial identities
O článku:
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.
Ako citovať:
ISO 690:
Ramírez, J., Shattuck, M. 2018. A multi-parameter generalization of the symmetric algorithm. In Mathematica Slovaca, vol. 68, no.4, pp. 699-712. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0137

APA:
Ramírez, J., Shattuck, M. (2018). A multi-parameter generalization of the symmetric algorithm. Mathematica Slovaca, 68(4), 699-712. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0137
O vydaní:
Publikované: 6. 8. 2018