Continuant (mathematics) explained

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 byK _{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 & … & 0 \ -1 & x _{2} & 1 & \ddots & \vdots\\ 0 & -1 & \ddots &\ddots & 0 \\ \vdots & \ddots & \ddots &\ddots & 1 \ 0 & … & 0 & -1 &x _{n \end{pmatrix}.} K _{n(1, \ldots, 1)} = F _{n+1} , the (n+1)-st Fibonacci number.(K _{n(x} _{1, \ldots, x} _{n)})/(K _{n-1} (x _{2, \ldots, x} _{n)}) = x _{1} + (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:

(K _{n(x} _{1, \ldots,x} _{n)})/(K _{n-1} (x _{2, \ldots, x} _{n)}) = [x _{1; x} _{2, \ldots, x} _{n]} = x _{1} + (1)/(\displaystyle{x _{2} + (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} x \ldots x \begin{pmatrix} x _{n} & 1 \ 1 & 0 \end{pmatrix} .

• • For determinants, it implies that

K _{n(x} _{1, \ldots, x} _{n) ⋅} K _{n-2} (x _{2, \ldots, x} _{n-1} ) - K _{n-1} (x _{1, \ldots, x} _{n-1} ) ⋅ K _{n-1} (x _{2, \ldots, x} _{n} ) = ( -1 ) ^{n.}

• • and also

K _{n-1} (x _{2, \ldots, x} _{n) ⋅} K _{n+2} (x _{1, \ldots, x} _{n+2} ) - K _{n(x} _{1, \ldots, x} _{n) ⋅} 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

• Book: Thomas Muir (mathematician)

. Thomas Muir . Thomas Muir (mathematician) . A treatise on the theory of determinants . registration . 1960 . . 516 - 525 .

• Book: The Markoff and Lagrange Spectra . Thomas W. . Cusick . Mary E. . Flahive . Mary Flahive . . 1989 . 0-8218-1531-8 . 89 . 0685.10023 . Mathematical Surveys and Monographs . 30 . Providence, RI .
• Book: George Chrystal

. Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1 . George Chrystal . George Chrystal . American Mathematical Society . 1999 . 0-8218-1649-7 . 500 .