Linear vector recursions of arbitrary order

Bernadette Faye; Laszlo Nemeth; Laszlo Szalay

doi:10.47443/dml.2024.029

Author(s)

Bernadette Faye Department of Mathematics, University Alioune Diop de Bambey,Bambey,Senegal

Laszlo Nemeth Department of Mathematics, Institute of Basic Sciences, University of Sopron,Sopron,Hungary; Department of Mathematics, J. Selye University,Kom a ́rno,Slovakia

Laszlo Szalay Department of Mathematics, Institute of Basic Sciences, University of Sopron,Sopron,Hungary; Department of Mathematics, J. Selye University,Kom a ́rno,Slovakia

Abstract

Solution of various combinatorial problems often requires vector recurrences of higher order (i.e., the order is larger than 1). Assume that there are given matrices A1; A2; : : : ; As, all from Ck k. These matrices allow us to define the vector recurrence vn = A1vn 1 + A2vn 2 + + Asvn s for the vectors vn 2 Ck, n s. The paramount result of this paper is that we could separate the component sequences of the vectors and find a common linear recurrence relation to describe them. The principal advantage of our approach is a uniform treatment and the possibility of automatism. We could apply the main result to answer a problem that arose concerning the rows of the modified hyperbolic Pascal triangle with parameters f4; 5g. We also verified two other statements from the literature in order to illustrate the power of the method.

Publication Details

Page(s) 50-57

DOI 10.47443/dml.2024.029

Published Journal: Discrete Mathematics Letters, Volume: 13, Issue: 0, Year: 2024

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Keywords

linear recurrence, vector recurrence, hyperbolic Pascal triangle, bounded anchored permutation, Fibonacci sequence

References

Atanassov K. T.,Atanassova L. C.,Shannon A. G.,Fibonacci sequences,Belbachir H.,M. M. Gillespie K. G.,Monks K. G.,K. M. Monks K. G.,Ne L.,Ne L.,Ne L.,N. J. A. Sloane, L.,H. C. A. van Tilborg L. .2005 .The On-Line Encyclopedia of Integer Sequences, OEIS Foundation Inc. Encyclopedia of Cryptography and Security, 343(2017) : 189-203.