In algebra, Brahmagupta polynomials are a class of polynomials associated with the Brahmagupta matrix, which in turn is associated with Brahmagupta's identity. The concept and terminology were introduced by E. R. Suryanarayan, University of Rhode Island, Kingston in a paper published in 1996. These polynomials have several interesting properties and have found applications in tiling problems(In the proceedings, see paper authored by R. Rangarajan and E. R. Suryanarayan and titled "The Brahmagupta Matrix and its applications") and in the problem of finding Heronian triangles in which the lengths of the sides are consecutive integers. They are named for the 7th-century Indian mathematician Brahmagupta.

Definition Brahmagupta's identity Brahmagupta's identity says that, for a given integer , the product of two numbers of the form is again a number of the form. More precisely, we have

This identity can be used to generate infinitely many solutions to Pell's equation. It can also be used to generate successively better rational approximations to square roots of arbitrary integers.

Brahmagupta matrix If, for an arbitrary real number , we define the matrix:

Then, Brahmagupta's identity can be expressed in the following form:

The matrix is called the Brahmagupta matrix.

Brahmagupta polynomials

Let be as above. Then, it can be seen by induction that the matrix can be written in the form

Here, and are polynomials in , , and . These polynomials are called the Brahmagupta polynomials. The first few of the polynomials are listed below:

Properties A few elementary properties of the Brahmagupta polynomials are summarized here. More advanced properties are discussed in the paper by Suryanarayan.

Recurrence relations The polynomials and satisfy the following recurrence relations:

Exact expressions

The eigenvalues of are and the corresponding eigenvectors are Hence

It follows that

This yields the following exact expressions for and :

Expanding the powers in the above exact expressions using the binomial theorem and simplifying one gets the following expressions for and :

Special cases

If and then, for : is the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... is the Lucas sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...

If we set and , then: which are the numerators of continued fraction convergents to . This is also the sequence of half Pell-Lucas numbers. which is the sequence of Pell numbers.

A differential equation and are polynomial solutions of the following partial differential equation:

References