Brahmagupta polynomials are a class of polynomials associated with the Brahmagupa matrix which in turn is associated with the Brahmagupta's identity.
The concept and terminology were introduced by E. R. Suryanarayan, University of Rhode Island, Kingston in a paper published in 1996.
[1][2][3] These polynomials have several interesting properties and have found applications in tiling problems[4] and in the problem of finding Heronian triangles in which the lengths of the sides are consecutive integers.
[5] In algebra, Brahmagupta's identity says that, for given integer N, 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 the Pell's equation.
It can also be used to generate successively better rational approximations to square roots of arbitrary integers.
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.
Then, it can be seen by induction that the matrix
The first few of the polynomials are listed below: A few elementary properties of the Brahmagupta polynomials are summarized here.
More advanced properties are discussed in the paper by Suryanarayan.
satisfy the following recurrence relations: The eigenvalues of
Hence It follows that This yields the following exact expressions for
: Expanding the powers in the above exact expressions using the binomial theorem and simplifying one gets the following expressions for
are polynomial solutions of the following partial differential equation: