Английская Википедия:Continuant (mathematics)

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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>K_n(x_1,\;x_2,\;\ldots,\;x_n)</math> is defined recursively by

<math> K_0 = 1 ; \, </math>

<math> K_1(x_1) = x_1 ; \, </math>

<math> 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>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>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>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>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>K_n(1,\;\ldots,\;1) = F_{n+1}</math>, the (n+1)-st Fibonacci number.
• <math>\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>\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>\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>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>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> K_0 = 1 ; \, </math>

<math> K_1 = a_1 ; \, </math>

<math> 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> \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

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book