Continuant (mathematics)

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

Definition

The n-th continuant is defined recursively by

Properties

• The continuant can be computed by taking the sum of all possible products of x₁,...,x_n, in which any number of disjoint pairs of consecutive terms are deleted (Euler's rule). For example,
• :

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

• The continuant can be computed as the determinant of a tridiagonal matrix:
• :

• , the (n+1)-st Fibonacci number.
• Ratios of continuants represent (convergents to) continued fractions as follows:
• :
• The following matrix identity holds:
• : .
• For determinants, it implies that
• :
• and also
• :

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,...,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 generalized continuant is precisely the determinant of the tridiagonal matrix

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

References