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,...,x_n, 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).

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

$K_n(1,\;\ldots,\;1) = F_{n+1}$ , 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:
$\begin{pmatrix} K_n(x_1,\;\ldots,\;x_n) & K_{n-1}(x_1,\;\ldots,\;x_{n-1}) \\ K_{n-1}(x_2,\;\ldots,\;x_n) & K_{n-2}(x_2,\;\ldots,\;x_{n-1}) \end{pmatrix} = \begin{pmatrix} x_1 & 1 \\ 1 & 0 \end{pmatrix}\times\ldots\times\begin{pmatrix} x_n & 1 \\ 1 & 0 \end{pmatrix}$ .
- 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,...,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.

The extended 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.

References

Thomas Muir (1960). A treatise on the theory of determinants. Dover Publications. pp. 516–525.
Cusick, Thomas W.; Flahive, Mary E. (1989). The Markoff and Lagrange Spectra. Mathematical Surveys and Monographs. 30. Providence, RI: American Mathematical Society. p. 89. ISBN 0-8218-1531-8. Zbl 0685.10023.
George Chrystal (1999). Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1. American Mathematical Society. p. 500. ISBN 0-8218-1649-7.

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