# Continuant (mathematics)

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*, *K*(*n*), of a sequence **a** = *a*_{1},...,*a*_{n},... is defined recursively by

It may also be obtained by taking the sum of all possible products of *a*_{1},...,*a*_{n} in which any pairs of consecutive terms are deleted.

An extended 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 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.

## Applications

The simple continuant gives the value of a continued fraction of the form . The *n*-th convergent is

The extended continuant is precisely the determinant of the tridiagonal matrix

## References

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