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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.

Definition

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The n-th continuant is defined recursively by

Properties

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  • The continuant 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,
It follows that continuants are invariant with respect to reversing the order of indeterminates:
  • The continuant can be computed as the determinant of a tridiagonal matrix:
  • , the (n+1)-st Fibonacci number.
  • Ratios of continuants represent (convergents to) continued fractions as follows:
  • The following matrix identity holds:
    .
    • For determinants, it implies that
    • and also

Generalizations

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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

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 generalized continuant is precisely the determinant of the tridiagonal matrix

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

References

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  • Thomas Muir (1960). A treatise on the theory of determinants. Dover Publications. pp. 516–525.
  • Cusick, Thomas W.; Flahive, Mary E. (1989). The Markoff and Lagrange Spectra. Mathematical Surveys and Monographs. Vol. 30. Providence, RI: American Mathematical Society. p. 89. ISBN 0-8218-1531-8. Zbl 0685.10023.
  • George Chrystal (1999). Algebra, an Elementary Text-book for the Higher Classes of Secondary Schools and for Colleges: Pt. 1. American Mathematical Society. p. 500. ISBN 0-8218-1649-7.