Continuant (mathematics)

In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.

The n-th continuant ${\displaystyle K_{n}(x_{1},\;x_{2},\;\ldots ,\;x_{n})}$ is defined recursively by

A generalized definition takes the continuant with respect to three sequences a, b and c, so that K(n) is a polynomial of a₁,...,a_n, b₁,...,b_n−1 and c₁,...,c_n−1. In this case the recurrence relation becomes

Since b_r and c_r enter into K only as a product b_rc_r there is no loss of generality in assuming that the b_r are all equal to 1.

The extended continuant is precisely the determinant of the tridiagonal matrix

In Muir's book the generalized continuant is simply called continuant.

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