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 = [画像:x^n+t(n; 2)x^(n-2)y^2+t^2(n; 4)x^(n-4)y^4+...]
(3)
The Brahmagupta polynomials satisfy
(partialx_n)/(partialx) = [画像:(partialy_n)/(partialy)=nx_(n-1)]
(5)
(partialx_n)/(partialy) = [画像: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 =
(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 PolynomialsExplore with Wolfram|Alpha
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References
Suryanarayan, E. R. "The Brahmagupta Polynomials." Fib. Quart. 34, 30-39, 1996.Referenced on Wolfram|Alpha
Brahmagupta PolynomialCite this as:
Weisstein, Eric W. "Brahmagupta Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BrahmaguptaPolynomial.html