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 , 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 . Finally, we prove some results about congruences satisfied by the matrix Qn(a, b).
Highlights
In [4], Peele and Stanica studied n × n matrices with the (i, j) entry the binomial coefficients i−1 j−1(matrix Ln) and i−1 n−j, respectively, and derived many interesting results on powers of these matrices.The matrix Ln was subdued, but curiously enough, closed forms for entries of powers of Rn, say Rne, were not found
We prove that powers of 4-netted matrices preserve the property of nettedness: the entries of the eth power satisfy δea(i,ej) = αea(i−e)1,j +βea(i−e)1,j−1 +γea(i,ej)−1, where the coefficients are all instances of the same sequence xe+1 = (β + δ)xe −xe−1
As we will see in our first result, this is not a singular phenomenon
Summary
We prove that powers of 4-netted matrices (the entries satisfy a four-term recurrence δai,j = αai−1,j + βai−1,j−1 + γai,j−1) preserve the property of nettedness: the entries of the eth power satisfy δea(i,ej) = αea(i−e)1,j +βea(i−e)1,j−1 +γea(i,ej)−1, where the coefficients are all instances of the same sequence xe+1 = (β + δ)xe − (βδ+αγ)xe−1. 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. We prove some results about congruences satisfied by the matrix Qn(a, b). 2000 Mathematics Subject Classification: 05A10, 11B39, 11B65, 11C20, 15A36
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
