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 [math]\displaystyle{ K_n(x_1,\;x_2,\;\ldots,\;x_n) }[/math] is defined recursively by

[math]\displaystyle{ K_0 = 1 ; \, }[/math]

[math]\displaystyle{ K_1(x_1) = x_1 ; \, }[/math]

[math]\displaystyle{ 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}) . \, }[/math]

Properties

• The continuant [math]\displaystyle{ K_n(x_1,\;x_2,\;\ldots,\;x_n) }[/math] 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,

  [math]\displaystyle{ 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. }[/math]

It follows that continuants are invariant with respect to reversing the order of indeterminates: [math]\displaystyle{ K_n(x_1,\;\ldots,\;x_n) = K_n(x_n,\;\ldots,\;x_1). }[/math]

• The continuant can be computed as the determinant of a tridiagonal matrix:

  [math]\displaystyle{ 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}. }[/math]

• [math]\displaystyle{ K_n(1,\;\ldots,\;1) = F_{n+1} }[/math], the (n+1)-st Fibonacci number.
• [math]\displaystyle{ \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)}. }[/math]
• Ratios of continuants represent (convergents to) continued fractions as follows:

  [math]\displaystyle{ \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}}}. }[/math]

• The following matrix identity holds:

  [math]\displaystyle{ \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} }[/math].

  • For determinants, it implies that

    [math]\displaystyle{ 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. }[/math]

  • and also

    [math]\displaystyle{ 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}. }[/math]

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

[math]\displaystyle{ K_0 = 1 ; \, }[/math]

[math]\displaystyle{ K_1 = a_1 ; \, }[/math]

[math]\displaystyle{ K_n = a_n K_{n-1} - b_{n-1}c_{n-1} K_{n-2} . \, }[/math]

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

[math]\displaystyle{ \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} . }[/math]

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. https://archive.org/details/treatiseontheory0000muir. 
• Cusick, Thomas W.; Flahive, Mary E. (1989). The Markoff and Lagrange Spectra. Mathematical Surveys and Monographs. 30. Providence, RI: American Mathematical Society. pp. 89. ISBN 0-8218-1531-8. 
• George Chrystal (1999). Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1. American Mathematical Society. pp. 500. ISBN 0-8218-1649-7.