Pantelimon Stănică

Copyright © 2003 Pantelimon Stănică. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove that powers of 4-netted matrices (the entries satisfy a four-term recurrence δai,j=αai−1,j+βai−1,j+γai,j−1) preserve the property of nettedness: the entries of the eth power satisfy δeai,j(e)=αeai−1,j(e)+βeai−1,j−1(e)+γeai,j−1(e), where the coefficients are all instances of the same sequence xe+1=(β+δ)xe−(βδ+αγ)xe−1. Also, we find a matrix Qn(a,b) and a vector v such that Qn(a,b)e⋅v acts as a shifting on the general second-order recurrence sequence with parameters a, b. The shifting action of Qn(a,b) generalizes the known property (0111)e⋅(1,0)t=(Fe−1,Fe)t. Finally, we prove some results about congruences satisfied by the matrix Qn(a,b).