In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, the continuant is a
multivariate polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
representing the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of a
tridiagonal matrix
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main di ...
and having applications in
generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values.
A ge ...
s.
Definition
The ''n''-th ''continuant''
is defined recursively by
:
:
:
Properties
*The continuant
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,
*:
:It follows that continuants are invariant with respect to reversing the order of indeterminates:
*The continuant can be computed as the
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of a
tridiagonal matrix
In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal/lower diagonal (the first diagonal below this), and the supradiagonal/upper diagonal (the first diagonal above the main di ...
:
*:
*
, the (''n''+1)-st
Fibonacci number
In mathematics, the Fibonacci numbers, commonly denoted , form a sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from ...
.
*
* Ratios of continuants represent (convergents to)
continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...
s 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 ''a''
1,...,''a''
''n'', ''b''
1,...,''b''
''n''−1 and ''c''
1,...,''c''
''n''−1. In this case the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
becomes
:
:
:
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
:
In Muir's book the generalized continuant is simply called continuant.
References
*
*
* {{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
Polynomials