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 $K_n(x_1,x_2,\dots ,x_n)$ is defined recursively by

$K_0=1;$

$K_1(x_1)=x_1;$

$K_n(x_1,x_2,\dots ,x_n)=x_n{K}_{n-1}(x_1,x_2,\dots ,x_{n-1})+K_{n-2}(x_1,x_2,\dots ,x_{n-2}).$

Properties

• The continuant $K_n(x_1,x_2,\dots ,x_n)$ 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: $K_n(x_1,\dots ,x_n)=K_n(x_n,\dots ,x_1).$

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

  $K_n(x_1,x_2,\dots ,x_n)=\det \left(\begin{array}{ccccc}{x}_1& 1& 0& \cdots & 0\\ -1& x_2& 1& \ddots & ⋮\\ 0& -1& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & 1\\ 0& \cdots & 0& -1& x_n\end{array}\right).$

• $K_n(1,\dots ,1)=F_{n+1}$, the (n+1)-st Fibonacci number.
• $\frac{K_n(x_1,\dots ,x_n)}{K_{n-1}(x_2,\dots ,x_n)}=x_1+\frac{K_{n-2}(x_3,\dots ,x_n)}{K_{n-1}(x_2,\dots ,x_n)}.$
• Ratios of continuants represent (convergents to) continued fractions as follows:

  $\frac{K_n(x_1,\dots ,x_n)}{K_{n-1}(x_2,\dots ,x_n)}=[x_1;x_2,\dots ,x_n]=x_1+\frac{1}{x_2+\frac{1}{x_3+\dots }}.$

• The following matrix identity holds:

  $\left(\begin{array}{cc}{K}_n(x_1,\dots ,x_n)& K_{n-1}(x_1,\dots ,x_{n-1})\\ K_{n-1}(x_2,\dots ,x_n)& K_{n-2}(x_2,\dots ,x_{n-1})\end{array}\right)=\left(\begin{array}{cc}{x}_1& 1\\ 1& 0\end{array}\right)\times \dots \times \left(\begin{array}{cc}{x}_n& 1\\ 1& 0\end{array}\right)$.

  • For determinants, it implies that

    $K_n(x_1,\dots ,x_n)\cdot K_{n-2}(x_2,\dots ,x_{n-1})-K_{n-1}(x_1,\dots ,x_{n-1})\cdot K_{n-1}(x_2,\dots ,x_n)=(-1{)}^n.$

  • and also

    $K_{n-1}(x_2,\dots ,x_n)\cdot K_{n+2}(x_1,\dots ,x_{n+2})-K_n(x_1,\dots ,x_n)\cdot K_{n+1}(x_2,\dots ,x_{n+2})=(-1{)}^{n+1}{x}_{n+2}.$

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

$\left(\begin{array}{cccccc}{a}_1& b_1& 0& \dots & 0& 0\\ c_1& a_2& b_2& \dots & 0& 0\\ 0& c_2& a_3& \dots & 0& 0\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& 0& \dots & a_{n-1}& b_{n-1}\\ 0& 0& 0& \dots & c_{n-1}& a_n\end{array}\right).$

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. Vol. 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.