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

:$K_0 = 1 ; \,$
:$K_1(x_1) = x_1 ; \,$
:$K_n(x_1,\;x_2,\;\ldots,\;x_n) = x_n K_(x_1,\;x_2,\;\ldots,\;x_) + K_(x_1,\;x_2,\;\ldots,\;x_) . \,$

Properties

*The continuant $K_n(x_1,\;x_2,\;\ldots,\;x_n)$ can be computed by taking the sum of all possible products of ''x''₁,...,''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:
*: $K_n(x_1,\;x_2,\;\ldots,\;x_n)= \det \begin x_1 & 1 & 0 &\cdots & 0 \\ -1 & x_2 & 1 & \ddots & \vdots\\ 0 & -1 & \ddots &\ddots & 0 \\ \vdots & \ddots & \ddots &\ddots & 1 \\ 0 & \cdots & 0 & -1 &x_n \end.$

* $K_n(1,\;\ldots,\;1) = F_$, the (''n''+1)-st Fibonacci number.
* $\frac = x_1 + \frac.$
* Ratios of continuants represent (convergents to) continued fractions as follows:
*: \frac = _1;\;x_2,\;\ldots,\;x_n= x_1 + \frac.
* The following matrix identity holds:
*: $\begin K_n(x_1,\;\ldots,\;x_n) & K_(x_1,\;\ldots,\;x_) \\ K_(x_2,\;\ldots,\;x_n) & K_(x_2,\;\ldots,\;x_) \end = \begin x_1 & 1 \\ 1 & 0 \end\times\ldots\times\begin x_n & 1 \\ 1 & 0 \end$.
**For determinants, it implies that
**:$K_n(x_1,\;\ldots,\;x_n)\cdot K_(x_2,\;\ldots,\;x_) - K_(x_1,\;\ldots,\;x_)\cdot K_(x_2,\;\ldots,\;x_) = (-1)^n.$
**and also
**:$K_(x_2,\;\ldots,\;x_n)\cdot K_(x_1,\;\ldots,\;x_) - K_n(x_1,\;\ldots,\;x_n)\cdot K_(x_2,\;\ldots,\;x_) = (-1)^ x_.$

Generalizations

A generalized definition takes the continuant with respect to three sequences a, b and c, so that ''K''(''n'') is a polynomial of ''a''₁,...,''a''_''n'', ''b''₁,...,''b''_''n''−1 and ''c''₁,...,''c''_''n''−1. In this case the recurrence relation becomes

:$K_0 = 1 ; \,$
:$K_1 = a_1 ; \,$
:$K_n = a_n K_ - b_c_ K_ . \,$

Since ''b''_''r'' and ''c''_''r'' enter into ''K'' only as a product ''b''_''r''''c''_''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

:$\begin a_1 & b_1 & 0 & \ldots & 0 & 0 \\ c_1 & a_2 & b_2 & \ldots & 0 & 0 \\ 0 & c_2 & a_3 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & a_ & b_ \\ 0 & 0 & 0 & \ldots & c_ & a_n \end .$

In Muir's book the generalized continuant is simply called continuant.

References

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* {{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

Continued fractions
Matrices (mathematics)
Polynomials