# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The combinatorial power of the companion matrix. (English) Zbl 0838.15015
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.

##### MSC:
 15B57 Hermitian, skew-Hermitian, and related matrices
Full Text:
##### References:
 [1] Beerends, R. J.: Chebyshev polynomials in several variables and the radical part of the Laplace-Beltrami operator. Trans. amer. Math. soc. 328, 779-814 (1991) · Zbl 0739.22008 [2] Bonetti, F.; Rota, G. -C.; Senato, D.; Venezia, A. M.: On the foundation of combinatorial theory, X, A categorical setting for symmetric functions. Stud. appl. Math. 86, 1-29 (1992) · Zbl 0748.05093 [3] Brualdi, R. A.; Ryser, H. J.: Combinatorial matrix theory. Encyclopedia math. Appl. 39 (1991) · Zbl 0746.05002 [4] L. Brutman, The Fourier operator of even order and its application to an extremum problem in interpolation, in Algorithms for Approximation 2 (J.C. Mason and M.G. Cox, Eds.), Chapman and Hall, London, pp. 170--176. · Zbl 0747.41001 [5] Chen, W. Y. C.: Context-free grammars, differential operators and formal power series. Theoret. comput. Sci. 117, 113-129 (1993) · Zbl 0788.68082 [6] Chen, W. Y. C.; Lih, K. W.; Yeh, Y. N.: Cyclic tableaux and symmetric functions. Stud. appl. Math. 94, 327-333 (1995) · Zbl 0824.05061 [7] Coddington, E. A.; Levinson, N.: Theory of ordinary differential equations. (1955) · Zbl 0064.33002 [8] Comtet, L.: Advanced combinatorics. (1974) · Zbl 0283.05001 [9] Doubilet, P.: On the foundations of combinatorial theory VII: Symmetric functions through the theory of distribution and occupancy. Stud. appl. Math. 51, 377-396 (1972) · Zbl 0274.05008 [10] Friedman, B.: Principles and techniques of applied mathematics. (1956) · Zbl 0072.12806 [11] Fromme, J. A.; Goldberg, M. A.: Equations arising in two-dimensional aerodynamics. Solution methods for integral equations (1979) · Zbl 0434.65102 [12] Gao, S.; Jr., H. W. Lenstra: Optimal normal bases. Des. codes cryptogr. 2, 315-323 (1992) [13] S. Gao and G. L. Mullen, Dickson polynomials and irreducible polynomials over finite fields, J. Number Theory, to appear. · Zbl 0810.11070 [14] Gautschi, W.: On mean convergence of extended Lagrange interpolation. J. comput. Appl. math. 43, 19-35 (1992) · Zbl 0761.41003 [15] Goulden, I. P.; Jackson, D. M.: Combinatorial enumeration. (1983) · Zbl 0519.05001 [16] R. Kit Kittappa, A representation of the solution of the nth order linear difference equation with variable coefficients, preprint. [17] Lancaster, P.; Tismenesky, M.: The theory of matrices. (1985) [18] Lidl, R.; Mullen, G. L.; Turnwald, G.: Dickson polynomials. Pitman monographs surveys pure appl. Math. 65 (1993) · Zbl 0823.11070 [19] Lidl, R.; Niederreiter, H.: Introduction to finite fields and their applications. (1986) · Zbl 0629.12016 [20] Louck, J. D.: Exact normal modes of oscillation of a linear chain of identical particles. Amer. J. Phys. 30, No. 8, 585-5900 (1962) · Zbl 0129.43805 [21] Louck, J. D.: Trace formulas for a rigid asymmetric rotator-type Hamiltonian. J. molecular spectroscopy 10, 263-277 (1963) [22] Macdonald, I. G.: Symmetric functions and Hall polynomials. (1979) · Zbl 0487.20007 [23] Mason, J. C.: Chebyshev polynomials of the second, third and fourth kinds in approximation, indefinite integration, and integral transforms. J. comput. Appl. math. 49, 169-178 (1993) · Zbl 0793.33010 [24] Mason, J. C.; Elliott, G. H.: Near-minimax complex approximation by four kinds of Chebyshev polynomial expansion. J. comput. Appl. math. 46, 291-300 (1993) · Zbl 0782.30031 [25] Mason, J. C.; Elliott, G. H.: Constrained near-minimax approximation by weighted expansion and interpolation using Chebyshev polynomials of the second, third and fourth kinds. Algorithms for constrained approximation and optimization, proc. Workshop univ. Vermont (1993) [26] Mullin, R. C.; Onyszchuk, I.; Vanstone, S. A.; Wilson, R. M.: Optimal normal bases in $GF(pn)$. Discrete applied math. 22, 149-161 (1988--1989) · Zbl 0661.12007 [27] Rota, G. -C.: Baxter algebras and combinatorial identities, parts II. Bull. amer. Math. soc. 75, 330-334 (1969) · Zbl 0319.05008 [28] Rybowicz, M.: Search of primitive polynomials over finite fields. J. pure and appl. Algebra 65, 139-151 (1990) · Zbl 0713.11085 [29] Schmitt, W. R.: Antipodes and incidence coalgebras. J. combin. Theory ser. A 46, 264-290 (1987) · Zbl 0699.05003 [30] Stanley, R. P.: Enumerative combinatorics. (1986) · Zbl 0608.05001