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 is defined recursively by

Properties

A generalized definition takes the continuant with respect to three sequences a, b and c, so that K is a polynomial of a1,...,an, b1,...,bn−1 and c1,...,cn−1. In this case the recurrence relation becomes
Since br and cr enter into K only as a product brcr there is no loss of generality in assuming that the br 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.