View source for Continuant

{{MSC|11Y65}}

An algebraic function of a sequence of variables which has applications to [[continued fraction]]s and as the determinant of a   [[tridiagonal matrix]].

The $n$-th ''continuant'', $K(n)$, of a sequence $\mathbf{a} = a_1,\ldots,a_n,\ldots$  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   $a_1,\ldots,a_n$ in which any pairs of consecutive terms are deleted.

An extended definition takes the continuant with respect to three   sequences $\mathbf a$, $\mathbf b$, $\mathbf c$, so that $K(n)$ is a   function of $a_1,\ldots,a_n$, $b_1,\ldots,b_{n-1}$,   $c_1,\ldots,c_{n-1}$.  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 the product $b_r c_r$ there is   no loss of generality in assuming that the $b_r$ are all equal to 1.

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

==References==
* Thomas Muir.  ''A treatise on the theory of determinants''. (Dover Publications, 1960 [1933]), pp.  516-525.