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

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

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

Taking $x=y=1$ and $t=2$ gives $y_n$ equal to the Pell Numbers and $x_n$ 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.


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

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© 1996-9 Eric W. Weisstein