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Zbl 0838.15015
Chen, William Y.C.; Louck, James D.
The combinatorial power of the companion matrix. (English)
[J] Linear Algebra Appl. 232, 261-278 (1996). ISSN 0024-3795

The explicit polynomials for all elements in an arbitrary power of the companion matrix depending on $n$ variables are obtained using combinatorial methods. Several applications are discussed as well as the relationship with Waring's formula on symmetric functions, the general solution to homogeneous linear recurrence relations, the multiplicative inverse of formal power series, the generating function of compositions (of numbers), a unified approach to Chebyshev polynomials, Dickson polynomials of various kinds arising from the theory of finite fields and combinatorial expansions of Toeplitz matrices.
[G.Bonanno (Davis)]

MSC 2000:: *15A57 Other types of matrices

Keywords: combinatorial power; companion matrix; Waring's formula; recurrence relations; formal power series; Chebyshev polynomials; Dickson polynomials; finite fields; Toeplitz matrices

Cited in: Zbl 1031.15019

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