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[edit]

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_n(x_1,\;x_2,\;\ldots,\;x_{n-1}) + K_{n-2}(x_1,\;x_2,\;\ldots,\;x_{n-2}) . \,

Properties[edit]

  • Continuant K_n(x_1,\;x_2,\;\ldots,\;x_n) can be computed by taking the sum of all possible products of x1,...,xn, 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 reverting the order of indeterminates: K_n(x_1,\;\ldots,\;x_n) = K_n(x_n,\;\ldots,\;x_1).
  • \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)}.
  • 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}\times\ldots\times\begin{pmatrix} x_n & 1 \\ 1 & 0 \end{pmatrix}.
    • For determinants, it implies that
      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.
    • and also
      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}.

Generalizations[edit]

An generalized definition takes the continuant with respect to three sequences a, b and c, so that K(n) is a polynomial of a1,...,an, b1,...,bn−1 and c1,...,cn−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 br and cr enter into K only as a product brcr there is no loss of generality in assuming that the br are all equal to 1.

The extended[citation needed] 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[edit]