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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
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