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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.

Contents

Definition

The n-th continuant K n ( x 1 , x 2 , , x n ) is defined recursively by

K 0 = 1 ; K 1 ( x 1 ) = x 1 ; K n ( x 1 , x 2 , , x n ) = x n K n 1 ( x 1 , x 2 , , x n 1 ) + K n 2 ( x 1 , x 2 , , x n 2 ) .

Properties

  • The continuant K n ( x 1 , x 2 , , 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 reversing the order of indeterminates: K n ( x 1 , , x n ) = K n ( x n , , x 1 ) .
  • The continuant can be computed as the determinant of a tridiagonal matrix:
  • K n ( 1 , , 1 ) = F n + 1 , the (n+1)-st Fibonacci number.
  • K n ( x 1 , , x n ) K n 1 ( x 2 , , x n ) = x 1 + K n 2 ( x 3 , , x n ) K n 1 ( x 2 , , x n ) .
  • Ratios of continuants represent (convergents to) continued fractions as follows:
  • The following matrix identity holds:
  • For determinants, it implies that
  • and also
  • 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,...,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 continuant is precisely the determinant of the tridiagonal matrix

    ( a 1 b 1 0 0 0 c 1 a 2 b 2 0 0 0 c 2 a 3 0 0 0 0 0 a n 1 b n 1 0 0 0 c n 1 a n ) .

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

    References

    Continuant (mathematics) Wikipedia


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