Verde-Star, Luis Polynomial sequences generated by infinite Hessenberg matrices. (English) Zbl 1360.15034 Spec. Matrices 5, 64-72 (2017). Summary: We show that an infinite lower Hessenberg matrix Wikipedia Wolfram MathWorld generates polynomial sequences Wikipedia Wolfram MathWorld that correspond to the rows of infinite lower triangular nLab Wikipedia Wikipedia Wolfram MathWorld Wolfram MathWorld invertible matrices Encyclopedia of Mathematics Wikipedia Wolfram MathWorld . Orthogonal polynomial sequences Wikipedia Wolfram MathWorld are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices Wikipedia Wolfram MathWorld , matrix similarity Wikipedia Wolfram MathWorld Wolfram MathWorld , construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.© 2017 Cited in 1 ReviewCited in 15 Documents MSC: 15B05 Toeplitz, Cauchy, and related matrices 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 15A21 Canonical forms, reductions, classification 05A15 Exact enumeration problems, generating functions Keywords:polynomial sequences of interpolatory type; infinite Toeplitz matrices; infinite Hessenberg matrices; Toeplitz companion matrices; orthogonal polynomial sequences; recurrence relations; matrix similarity; generating functions × Cite Format Result Cite Review PDF Full Text: DOI References: [1] A. Böttcher, S. M. Grudsky, Toeplitz Matrices, Asymptotic Linear Algebra and Functional Analysis, Birkhäuser, Basel, 2000.; · Zbl 0952.47025 [2] A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices, Springer, New York, 1999.; · Zbl 0916.15012 [3] D. S. Mackey, N. Mackey, S. Petrovic, Is every matrix similar to a Toeplitz matrix?, Linear Algebra Appl. 297 (1999) 87-105.; · Zbl 0939.15003 [4] M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices, 2013; 1:10-16.; · Zbl 1291.15015 [5] L. Verde-Star, Infinite triangular matrices, q-Pascal matrices, and determinantal representations, Linear Algebra Appl. 434 (2011) 307-318.; · Zbl 1203.15021 [6] L. Verde-Star, Characterization and construction of classical orthogonal polynomials using a matrix approach, Linear Algebra Appl. 438 (2013) 3635-3648.; · Zbl 1275.33022 [7] L. Verde-Star, Polynomial sequences of interpolatory type, Studies Appl. Math. 88 (1993) 153-172.; · Zbl 0774.41007 [8] L. Verde-Star, Elementary triangular matrices and inverses of k-Hessenberg matrices and triangular matrices, Spec. Matrices, 2015; 3: 250-256.; · Zbl 1329.15073 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.