Continuant (mathematics)

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

References

{{cite book | author=Thomas Muir | authorlink=Thomas Muir (mathematician) | title=A treatise on the theory of determinants | url=https://archive.org/details/treatiseontheory0000muir | url-access=registration | date=1960 | publisher=Dover Publications | pages=[https://archive.org/details/treatiseontheory0000muir/page/516 516]–525 }}
{{cite book | title=The Markoff and Lagrange Spectra | first1=Thomas W. | last1=Cusick | first2=Mary E. | last2=Flahive |author2-link= Mary Flahive | publisher=American Mathematical Society | year=1989 | isbn=0-8218-1531-8 | pages=89 | zbl=0685.10023 | series=Mathematical Surveys and Monographs | volume=30 | location=Providence, RI }}
{{cite book | title=Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1 | author=George Chrystal | authorlink=George Chrystal | publisher=American Mathematical Society | year=1999 | isbn=0-8218-1649-7 | pages=500 }}

Category:Continued fractions

Category:Matrices (mathematics)

Category:Polynomials