Companion matrices turn log-concavity of linear-recurrence sequences with linear coefficients into a checkable positive-semidefinite condition, with tight infinite-log-concavity criteria for second-order constant-coefficient cases.
Brenti, Combinatorics and total positivity, J
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Log-Concavity and Infinite Log-Concavity of Linear Recurrent Sequences with Linear Coefficients via Companion Matrix Methods
Companion matrices turn log-concavity of linear-recurrence sequences with linear coefficients into a checkable positive-semidefinite condition, with tight infinite-log-concavity criteria for second-order constant-coefficient cases.