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Brahmagupta Polynomial


One of the polynomials obtained by taking powers of the Brahmagupta matrix. They satisfy the recurrence relation

x_(n+1)=xx_n+tyy_n
(1)
y_(n+1)=xy_n+yx_n.
(2)

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

x_n=x^n+t(n; 2)x^(n-2)y^2+t^2(n; 4)x^(n-4)y^4+...
(3)
y_n=nx^(n-1)y+t(n; 3)x^(n-3)y^3+t^2(n; 5)x^(n-5)y^5+....
(4)

The Brahmagupta polynomials satisfy

(partialx_n)/(partialx)=(partialy_n)/(partialy)=nx_(n-1)
(5)
(partialx_n)/(partialy)=t(partialy_n)/(partialy)=nty_(n-1).
(6)

The first few polynomials are

x_0=1
(7)
x_1=x
(8)
x_2=x^2+ty^2
(9)
x_3=x^3+3txy^2
(10)
x_4=x^4+6tx^2y^2+t^2y^4
(11)

and

y_0=0
(12)
y_1=y
(13)
y_2=2xy
(14)
y_3=3x^2y+ty^3
(15)
y_4=4x^3y+4txy^3.
(16)

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.


See also

Morgan-Voyce Polynomials

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References

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

Referenced on Wolfram|Alpha

Brahmagupta Polynomial

Cite this as:

Weisstein, Eric W. "Brahmagupta Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrahmaguptaPolynomial.html

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