Netted matrices

Abstract

We prove that powers of 4-netted matries (the entries satisfy a four-term recurrence daij = aai-Ij+Bai-ij+yaij-1) preserve this property of nettedness, that is the entries of the e-th power satisfy {equation} where the coefficients are all instances of tha same sequence, {equation}. Also, we find a matrix Qn(a,b) and a vector v, such that Qn(a,b)2 -v acts as a shifting on the general second-order recurrence sequence with parameters a,b. It generalizes the known property {equation}. We use it to prove a few interesting identities. In the last section we prove some results about congruences satisfied by the matrix Qn(a,b).