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,...,a_n,... 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,...,a_n 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,...,a_n, b₁,...,b_n-1 and c₁,...,c_n-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 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.

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

${\begin{pmatrix}a_{1}&b_{1}&0&0&\ldots &0&0\\c_{1}&a_{2}&b_{2}&0&\ldots &0&0\\0&c_{2}&a_{3}&b_{3}&\ldots &0&0\\\vdots &\ddots &\ddots &\ddots &&\vdots &\vdots \\0&0&0&0&\ldots &c_{n-1}&a_{n}\end{pmatrix}}.$