Brahmagupta polynomials

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 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 Template:Mvar, the product of two numbers of the form Template:Math is again a number of the form. More precisely, we have $$(x_1^2-N{y}_1^2)(x_2^2-N{y}_2^2)=(x_1{x}_2+N{y}_1{y}_2{)}^2-N(x_1{y}_2+x_2{y}_1{)}^2.$$ 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 Template:Mvar, we define the matrix: $$B(x,y)=\left[\begin{array}{cc}x& y\\ ty& x\end{array}\right]$$ Then, Brahmagupta's identity can be expressed in the following form: $$detB(x_1,y_1)detB(x_2,y_2)=det(B(x_1,y_1)B(x_2,y_2))$$ The matrix Template:Math is called the Brahmagupta matrix.

Brahmagupta polynomials

Let Template:Math be as above. Then, it can be seen by induction that the matrix Template:Mvar can be written in the form $$B^n=\left[\begin{array}{cc}{x}_n& y_n\\ t{y}_n& x_n\end{array}\right]$$ Here, Template:Mvar and Template:Mvar are polynomials in Template:Mvar, Template:Mvar, and Template:Mvar. These polynomials are called the Brahmagupta polynomials. The first few of the polynomials are listed below:

$$\begin{array}{cccc}{x}_1& =x& y_1& =y\\ x_2& =x^2+t{y}^2& y_2& =2xy\\ x_3& =x^3+3tx{y}^2& y_3& =3x^{2}y+t{y}^3\\ x_4& =x^4+6t^2{x}^2{y}^2+t^2{y}^4& y_4& =4x^{3}y+4tx{y}^3\end{array}$$

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 Template:Mvar and Template:Mvar satisfy the following recurrence relations: $$\begin{array}{cc}{x}_{n+1}& =x{x}_n+ty{y}_n\\ y_{n+1}& =x{y}_n+y{x}_n\\ x_{n+1}& =2x{x}_n-(x^2-t{y}^2)x_{n-1}\\ y_{n+1}& =2x{y}_n-(x^2-t{y}^2)y_{n-1}\\ x_{2n}& =x_n^2+t{y}_n^2\\ y_{2n}& =2x_n{y}_n\end{array}$$

Exact expressions

The eigenvalues of Template:Math are $x\pm y\sqrt{t}$ and the corresponding eigenvectors are $[1,\pm \sqrt{t}{]}^{\mathrm{T}}.$ Hence $$B[1,\pm \sqrt{t}{]}^{\mathrm{T}}=(x\pm y\sqrt{t})[1,\pm \sqrt{t}{]}^{\mathrm{T}}.$$ It follows that $$B^n[1,\pm \sqrt{t}{]}^{\mathrm{T}}=(x\pm y\sqrt{t}{)}^n[1,\pm \sqrt{t}{]}^{\mathrm{T}}.$$ This yields the following exact expressions for Template:Mvar and Template:Mvar: $$\begin{array}{cc}{x}_n& =\frac{1}{2}[(x+y\sqrt{t}{)}^n+(x-y\sqrt{t}{)}^n]\\ y_n& =\frac{1}{2\sqrt{t}}[(x+y\sqrt{t}{)}^n-(x-y\sqrt{t}{)}^n]\end{array}$$

Expanding the powers in the above exact expressions using the binomial theorem and simplifying one gets the following expressions for Template:Mvar and Template:Mvar: $$\begin{array}{cc}{x}_n& =x^n+t(\genfrac{}{}{0ex}{}{n}{2})x^{n-2}{y}^2+t^2(\genfrac{}{}{0ex}{}{n}{4})x^{n-4}{y}^4+\cdots \\ y_n& =n{x}^{n-1}y+t(\genfrac{}{}{0ex}{}{n}{3})x^{n-3}{y}^3+t^2(\genfrac{}{}{0ex}{}{n}{5})x^{n-5}{y}^5+\cdots \end{array}$$

Special cases

1. If Template:Math and Template:Math then, for Template:Math:$$2y_n=F_n$$ is the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...$$2x_n=L_n$$ is the Lucas sequence 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ...

1. If we set Template:Math and Template:Math, then:$$x_n=1,1,3,7,17,41,99,239,577,\dots$$ which are the numerators of continued fraction convergents to $\sqrt{2}$. This is also the sequence of half Pell-Lucas numbers.$$y_n=0,1,2,5,12,29,70,169,408,\dots$$ which is the sequence of Pell numbers.

A differential equation

Template:Mvar and Template:Mvar are polynomial solutions of the following partial differential equation: $$(\frac{{\partial }^2}{\partial x^2}-\frac{1}{t}\frac{{\partial }^2}{\partial y^2})U=0$$