Continuant (mathematics)

Last updated

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

Contents

Definition

The n-th continuant is defined recursively by

Properties

It follows that continuants are invariant with respect to reversing the order of indeterminates:

Generalizations

A generalized definition takes the continuant with respect to three sequences a, b and c, so that K(n) is a polynomial of a1,...,an, b1,...,bn1 and c1,...,cn1. 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 generalized continuant is precisely the determinant of the tridiagonal matrix

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

References