## Netted matrices.(English)Zbl 1059.11019

Summary: We prove that powers of $$4$$-netted matrices (the entries satisfy a four-term recurrence $$\delta a_{i,j}=\alpha a_{i-1,j}+\beta a_{i-1,j-1}+\gamma a_{i,j-1}$$) preserve the property of nettedness: the entries of the $$e$$th power satisfy $$\delta_{e} a_{i,j}^{(e)}=\alpha_{e} a_{i-1,j}^{(e)}+ \beta_{e} a_{i-1,j-1}^{(e)}+\gamma_{e} a_{i,j-1}^{(e)}$$, where the coefficients are all instances of the same sequence $$x_{e+1}=(\beta+\delta)x_{e}-(\beta\delta+\alpha\gamma) x_{e-1}$$. Also, we find a matrix $$Q_{n}(a,b)$$ and a vector $$v$$ such that $$Q_{n}(a,b)^{e}\cdot v$$ acts as a shifting on the general second-order recurrence sequence with parameters $$a$$, $$b$$. The shifting action of $$Q_{n}(a,b)$$ generalizes the known property $$\left(\begin{smallmatrix} 0&1\𝟙&1 \end{smallmatrix}\right)^{e}\cdot(1,0)^t=(F_{e-1},F_{e})^t$$. Finally, we prove some results about congruences satisfied by the matrix $$Q_{n}(a,b)$$.

### MSC:

 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 11B65 Binomial coefficients; factorials; $$q$$-identities 11C20 Matrices, determinants in number theory 05A10 Factorials, binomial coefficients, combinatorial functions 15B36 Matrices of integers
Full Text: