Brahmagupta polynomials

In algebra, Brahmagupta's identity says that, for given integer N, the product of two numbers of the form $x^2\; -Ny^2$ is again a number of the form. More precisely, we have

:$(x\_1^2\; -\; Ny\_1^2)(x\_2^2\; -\; Ny\_2^2)\; =\; (x\_1x\_2\; +\; Ny\_1y\_2)^2\; -\; N(x\_1y\_2\; +\; x\_2y\_1)^2.$

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.

If, for an arbitrary real number $t$, we define the matrix

:

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

:$\backslash det\; B(x\_1,y\_1)\; \backslash det\; B(x\_2,y\_2)\; =\; \backslash det\; (\; B(x\_1,y\_1)B(x\_2,y\_2))$

The matrix $B(x,y)$ is called the Brahmagupta matrix.

Let $B=B(x,y)$ be as above. Then, it can be seen by induction that the matrix $B^n$ can be written in the form

:

Here, $x\_n$ and $y\_n$ are polynomials in $x,\; y,\; t$. These polynomials are called the Brahmagupta polynomials. The first few of the polynomials are listed below:

:$$

\begin{alignat}{2}

x_1 & = x & y_1 & = y \\

x_2 & = x^2+ty^2 & y_2 & = 2xy \\

x_3 & = x^3+3txy^2 & y_3 & = 3x^2y+ty^3 \\

x_4 & = x^4+6t^2x^2y^2+t^2y^4\qquad & y_4 & = 4x^3y +4txy^3

\end{alignat}

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

The polynomials $x\_n$ and $y\_n$ satisfy the following recurrence relations:

• $x\_\{n+1\}\; =\; xx\_n+tyy\_n$
• $y\_\{n+1\}=xy\_n+yx\_n$
• $x\_\{n+1\}\; =\; 2xx\_n\; -\; (x^2-ty^2)x\_\{n-1\}$
• $y\_\{n+1\}\; =\; 2xy\_n\; -\; (x^2-ty^2)y\_\{n-1\}$
• $x\_\{2n\}=x\_n^2+ty\_n^2$
• $y\_\{2n\}=2x\_ny\_n$

The eigenvalues of $B(x,y)$ are $x\backslash pm\; y\backslash sqrt\{t\}$ and the corresponding eigenvectors are $[1,\; \backslash pm\; \backslash sqrt\{t\}]^T$. Hence

:$B[1,\; \backslash pm\; \backslash sqrt\{t\}]^T\; =\; (x\backslash pm\; y\backslash sqrt\{t\})[1,\; \backslash pm\; \backslash sqrt\{t\}]^T$.

It follows that

:$B^n[1,\; \backslash pm\; \backslash sqrt\{t\}]^T\; =\; (x\backslash pm\; y\backslash sqrt\{t\})^n[1,\; \backslash pm\; \backslash sqrt\{t\}]^T$.

This yields the following exact expressions for $x\_n$ and $y\_n$:

• $x\_n\; =\; \backslash tfrac\{1\}\{2\}\backslash big[\; (x\; +\; y\backslash sqrt\{t\})^n\; +\; (x\; -\; y\backslash sqrt\{t\})^n\backslash big]$
• $y\_n\; =\; \backslash tfrac\{1\}\{2\backslash sqrt\{t\}\}\backslash big[\; (x\; +\; y\backslash sqrt\{t\})^n\; -\; (x\; -\; y\backslash sqrt\{t\})^n\backslash big]$

Expanding the powers in the above exact expressions using the binomial theorem and simplifying one gets the following expressions for $x\_n$ and $y\_n$:

• $x\_n\; =\; x^n\; +t\; \{n\; \backslash choose\; 2\}\; x^\{n-2\}y^2\; +\; t^2\; \{n\backslash choose\; 4\}x^\{n-4\}y^4+\backslash cdots$
• $y\_n\; =\; nx^\{n-1\}y\; +t\{n\backslash choose\; 3\}x^\{n-3\}y^3\; +\; t^2\{n\; \backslash choose\; 5\}x^\{n-5\}y^5\; +\backslash cdots$

1. If $x=y=\backslash tfrac\{1\}\{2\}$ and $t=5$ then, for $n>0$:

:::$2y\_n=F\_n$ is the Fibonacci sequence $1,\; 1,\; 2,\; 3,\; 5,\; 8,\; 13,\; 21,\; 34,\; 55,\; \backslash ldots$.

:::$2x\_n=L\_n$ is the Lucas sequence $2,\; 1,\; 3,\; 4,\; 7,\; 11,\; 18,\; 29,\; 47,\; 76,\; 123,\; \backslash ldots$.

1. If we set $x=y=1$ and $t=2$, then:

:::$x\_n=1,1,3,7,17,41,99,239,577,\backslash ldots$ which are the numerators of continued fraction convergents to $\backslash sqrt\{2\}$.{{cite web |last1=N. J. A. Sloane |title=A001333 |url=https://oeis.org/A001333 |website=The On-Line Encyclopedia of Integer Sequences |access-date=1 December 2023}} This is also the sequence of half Pell-Lucas numbers.

:::$y\_n=\; 0,1,2,5,12,29,70,169,408,\; \backslash ldots$ which is the sequence of Pell numbers.

$x\_n$ and $y\_n$ are polynomial solutions of the following partial differential equation:

::$\backslash left(\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; -\; \backslash frac\{1\}\{t\}\backslash frac\{\backslash partial^2\}\{\backslash partial\; y^2\}\backslash right)U=0$

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Category:Polynomials

Category:Homogeneous polynomials

Category:Brahmagupta

Category:Matrices (mathematics)