Continuant (mathematics)

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

Definition
The n-th continuant $$K_n(x_1,\;x_2,\;\ldots,\;x_n)$$ is defined recursively by


 * $$ K_0 = 1 ; \, $$
 * $$ K_1(x_1) = x_1 ; \, $$
 * $$ K_n(x_1,\;x_2,\;\ldots,\;x_n) = x_n K_{n-1}(x_1,\;x_2,\;\ldots,\;x_{n-1}) + K_{n-2}(x_1,\;x_2,\;\ldots,\;x_{n-2}) . \, $$

Properties

 * The continuant $$K_n(x_1,\;x_2,\;\ldots,\;x_n)$$ can be computed by taking the sum of all possible products of x1,...,xn, in which any number of disjoint pairs of consecutive terms are deleted (Euler's rule). For example,
 * $$K_5(x_1,\;x_2,\;x_3,\;x_4,\;x_5) = x_1 x_2 x_3 x_4 x_5\; +\; x_3 x_4 x_5\; +\; x_1 x_4 x_5\; +\; x_1 x_2 x_5\; +\; x_1 x_2 x_3\; +\; x_1\; +\; x_3\; +\; x_5.$$
 * It follows that continuants are invariant with respect to reversing the order of indeterminates: $$K_n(x_1,\;\ldots,\;x_n) = K_n(x_n,\;\ldots,\;x_1).$$

\det \begin{pmatrix} x_1 & 1   & 0 &\cdots & 0 \\ -1 & x_2  & 1 &  \ddots    & \vdots\\ 0  & -1   & \ddots &\ddots & 0 \\ \vdots & \ddots & \ddots   &\ddots & 1 \\ 0 & \cdots & 0 & -1 &x_n \end{pmatrix}.$$
 * The continuant can be computed as the determinant of a tridiagonal matrix:
 * $$K_n(x_1,\;x_2,\;\ldots,\;x_n)=

\begin{pmatrix} x_1 & 1 \\ 1 & 0 \end{pmatrix}\times\ldots\times\begin{pmatrix} x_n & 1 \\ 1 & 0 \end{pmatrix}$$.
 * $$K_n(1,\;\ldots,\;1) = F_{n+1}$$, the (n+1)-st Fibonacci number.
 * $$\frac{K_n(x_1,\;\ldots,\;x_n)}{K_{n-1}(x_2,\;\ldots,\;x_n)} = x_1 + \frac{K_{n-2}(x_3,\;\ldots,\;x_n)}{K_{n-1}(x_2,\;\ldots,\;x_n)}.$$
 * Ratios of continuants represent (convergents to) continued fractions as follows:
 * $$\frac{K_n(x_1,\;\ldots,x_n)}{K_{n-1}(x_2,\;\ldots,\;x_n)} = [x_1;\;x_2,\;\ldots,\;x_n] = x_1 + \frac{1}{\displaystyle{x_2 + \frac{1}{x_3 + \ldots}}}.$$
 * The following matrix identity holds:
 * $$\begin{pmatrix} K_n(x_1,\;\ldots,\;x_n) & K_{n-1}(x_1,\;\ldots,\;x_{n-1}) \\ K_{n-1}(x_2,\;\ldots,\;x_n) & K_{n-2}(x_2,\;\ldots,\;x_{n-1}) \end{pmatrix} =
 * For determinants, it implies that
 * $$K_n(x_1,\;\ldots,\;x_n)\cdot K_{n-2}(x_2,\;\ldots,\;x_{n-1}) - K_{n-1}(x_1,\;\ldots,\;x_{n-1})\cdot K_{n-1}(x_2,\;\ldots,\;x_{n}) = (-1)^n.$$
 * and also
 * $$K_{n-1}(x_2,\;\ldots,\;x_n)\cdot K_{n+2}(x_1,\;\ldots,\;x_{n+2}) - K_n(x_1,\;\ldots,\;x_n)\cdot K_{n+1}(x_2,\;\ldots,\;x_{n+2}) = (-1)^{n+1} x_{n+2}.$$

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,...,bn&minus;1 and c1,...,cn&minus;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.

The generalized continuant is precisely the determinant of the tridiagonal matrix


 * $$ \begin{pmatrix}

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

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