Brahmagupta Polynomial

One of the Polynomials obtained by taking Powers of the Brahmagupta Matrix. They satisfy the recurrence relation

$\displaystyle x_{n+1}$	$\textstyle =$	$\displaystyle xx_n+tyy_n$	(1)
$\displaystyle y_{n+1}$	$\textstyle =$	$\displaystyle xy_n+yx_n.$	(2)

A list of many others is given by Suryanarayan (1996). Explicitly,

$\displaystyle x_n$	$\textstyle =$	$\displaystyle x^n+t{n\choose 2} x^{n-2}y^2+t^2{n\choose 4}x^{n-4}y^4+\ldots$	(3)
$\displaystyle y_n$	$\textstyle =$	$\displaystyle nx^{n-1}y+t{n\choose 3}x^{n-3}y^3+t^2{n\choose 5}x^{n-5}y^5+\ldots.$
			(4)

The Brahmagupta Polynomials satisfy

$\displaystyle {\partial x_n\over\partial x}$	$\textstyle =$	$\displaystyle {\partial y_n\over\partial y}=nx_{n-1}$	(5)
$\displaystyle {\partial x_n\over\partial y}$	$\textstyle =$	$\displaystyle t{\partial y_n\over\partial y}=nty_{n-1}.$	(6)

The first few Polynomials are

$\displaystyle x_0$	$\textstyle =$	$\displaystyle 0$
$\displaystyle x_1$	$\textstyle =$	$\displaystyle x$
$\displaystyle x_2$	$\textstyle =$	$\displaystyle x^2+ty^2$
$\displaystyle x_3$	$\textstyle =$	$\displaystyle x^3+3txy^2$
$\displaystyle x_4$	$\textstyle =$	$\displaystyle x^4+6tx^2y^2+t^2y^4$

and

$\displaystyle y_0$	$\textstyle =$	$\displaystyle 0$
$\displaystyle y_1$	$\textstyle =$	$\displaystyle y$
$\displaystyle y_2$	$\textstyle =$	$\displaystyle 2xy$
$\displaystyle y_3$	$\textstyle =$	$\displaystyle 3x^2y+ty^3$
$\displaystyle y_4$	$\textstyle =$	$\displaystyle 4x^3y+4txy^3.$

Taking

and

gives

equal to the Pell Numbers and

equal to half the Pell-Lucas numbers. The Brahmagupta Polynomials are related to the Morgan-Voyce Polynomials, but the relationship given by Suryanarayan (1996) is incorrect.

References

Suryanarayan, E. R. ``The Brahmagupta Polynomials.'' Fib. Quart. 34, 30-39, 1996.