International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 39, Pages 2507-2518

Netted matrices

Pantelimon Stănică

Department of Mathematics, Auburn University Montgomery, Montgomery 36124-4023, AL, USA

Received 11 January 2002

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.


We prove that powers of 4-netted matrices (the entries satisfy a four-term recurrence δai,j=αai1,j+βai1,j+γai,j1) preserve the property of nettedness: the entries of the eth power satisfy δeai,j(e)=αeai1,j(e)+βeai1,j1(e)+γeai,j1(e), where the coefficients are all instances of the same sequence xe+1=(β+δ)xe(βδ+αγ)xe1. Also, we find a matrix Qn(a,b) and a vector v such that Qn(a,b)ev 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=(Fe1,Fe)t. Finally, we prove some results about congruences satisfied by the matrix Qn(a,b).