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. These polynomials have several interesting properties and have found applications in tiling problems and in the problem of finding Heronian triangles in which the lengths of the sides are consecutive integers. 
Definition.
Brahmagupta's identity.
In algebra, Brahmagupta's identity says that, for given integer N, the product of two numbers of the form formula_1 is again a number of the form. More precisely, we have
formula_2
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.
Brahmagupta matrix.
If, for an arbitrary real number formula_3, we define the matrix
formula_4
then, Brahmagupta's identity can be expressed in the following form:
formula_5
The matrix formula_6 is called the Brahmagupta matrix.
Brahmagupta polynomials.
Let formula_7 be as above. Then, it can be seen by induction that the matrix formula_8 can be written in the form 
formula_9
Here, formula_10 and formula_11 are polynomials in formula_12. These polynomials are called the Brahmagupta polynomials. The first few of the polynomials are listed below:
formula_13
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 formula_10 and formula_11 satisfy the following recurrence relations:
Exact expressions.
The eigenvalues of formula_6 are formula_23 and the corresponding eigenvectors are formula_24. Hence
formula_25.
It follows that
formula_26.
This yields the following exact expressions for formula_10 and formula_11:
Expanding the powers in the above exact expressions using the binomial theorem and simplifying one gets the following expressions for formula_10 and formula_11:
formula_38 is the Fibonacci sequence formula_39. 
formula_40 is the Lucas sequence formula_41.
formula_44 which are the numerators of continued fraction convergents to formula_45. This is also the sequence of half Pell-Lucas numbers.
formula_46 which is the sequence of Pell numbers.
A differential equation.
formula_10 and formula_11 are polynomial solutions of the following partial differential equation:
formula_49
References.
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