Continuant (mathematics)

In algebra, the continuant of a sequence of terms is an algebraic expression which has applications in generalized continued fractions and as the determinant of a tridiagonal matrix.

Definition
The n-th continuant, K(n), of a sequence a = a1,...,an,... is defined recursively by


 * $$ K(0) = 1 ; \, $$
 * $$ K(1) = a_1 ; \, $$
 * $$ K(n) = a_n K(n-1) + K(n-2) . \, $$

It may also be obtained by taking the sum of all possible products of a1,...,an in which any pairs of consecutive terms are deleted.

An extended definition takes the continuant with respect to three sequences a, b and c, so that K(n) is a function of a1,...,an, b1,...,bn-1 and c1,...,cn-1. In this case the recurrence relation becomes


 * $$ K(0) = 1 ; \, $$
 * $$ K(1) = a_1 ; \, $$
 * $$ K(n) = a_n K(n-1) - b_{n-1}c_{n-1} K(n-2) . \, $$

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.

Applications
The simple continuant gives the value of a continued fraction of the form $$[a_0;a_1,a_2,\ldots]$$. The n-th convergent is


 * $$ \frac{K(n+1,(a_0,\ldots,a_n))}{K(n,(a_1,\ldots,a_n))} . $$

The extended continuant is precisely the determinant of the tridiagonal matrix


 * $$ \begin{pmatrix}

a_1 & b_1 & 0  &  0  & \ldots & 0 & 0 \\ c_1 & a_2 & b_2 & 0  & \ldots & 0 & 0 \\ 0 & c_2 & a_3 & b_3 & \ldots & 0 & 0 \\ \vdots & \ddots & \ddots & \ddots & & \vdots & \vdots \\ 0 & 0 & 0 & 0 & \ldots & c_{n-1} & a_n \end{pmatrix}. $$