Continuant (mathematics)

In algebra, the continuant is a multivariate polynomial representing the determinant of a tridiagonal matrix and having applications in generalized continued fractions.
Definition.
The "n"-th "continuant" formula_1 is defined recursively by
Generalizations.
A generalized definition takes the continuant with respect to three sequences a, b and c, so that "K"("n") is a polynomial of "a"1...,"a""n", "b"1...,"b""n"−1 and "c"1...,"c""n"−1. In this case the recurrence relation becomes
Since "b""r" and "c""r" enter into "K" only as a product "b""r""c""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.