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Morgan-Voyce Polynomials

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Morgan-VoycePolynomials

The Morgan-Voyce polynomials are polynomials related to the Brahmagupta and Fibonacci polynomials. They are defined by the recurrence relations

b_n(x)=xB_(n-1)(x)+b_(n-1)(x)
(1)
B_n(x)=(x+1)B_(n-1)(x)+b_(n-1)(x)
(2)

for n>=1, with

b_0(x)=B_0(x)=1.
(3)

Alternative recurrences are

b_n(x)=(x+2)b_(n-1)(x)-b_(n-2)(x)
(4)
B_n(x)=(x+2)B_(n-1)(x)-B_(n-2)(x)
(5)

with b_1(x)=1+x and B_1(x)=2+x, and

b_(n+1)b_(n-1)-b_n^2=x
(6)
B_(n+1)B_(n-1)-B_n^2=-1.
(7)

The polynomials can be given explicitly by the sums

B_n(x)=sum_(k=0)^(n)(n+k+1; n-k)x^k
(8)
b_n(x)=sum_(k=0)^(n)(n+k; n-k)x^k.
(9)

Defining the matrix

Q=[x+2 -1; 1 0]
(10)

gives the identities

Q^n=[B_n -B_(n-1); B_(n-1) -B_(n-2)]
(11)
Q^n-Q^(n-1)=[b_n -b_(n-1); b_(n-1) -b_(n-2)].
(12)

Defining

costheta=1/2(x+2)
(13)
coshphi=1/2(x+2)
(14)

gives

B_n(x)=(sin[(n+1)theta])/(sintheta)
(15)
B_n(x)=(sinh[(n+1)phi])/(sinhphi)
(16)

and

b_n(x)=(cos[1/2(2n+1)theta])/(cos(1/2theta))
(17)
b_n(x)=(cosh[1/2(2n+1)phi])/(cosh(1/2theta)).
(18)

The Morgan-Voyce polynomials are related to the Fibonacci polynomials F_n(x) by

b_n(x^2)=F_(2n+1)(x)
(19)
B_n(x^2)=1/xF_(2n+2)(x)
(20)

(Swamy 1968ab).

B_n(x) satisfies the ordinary differential equation

x(x+4)y^('')+3(x+2)y^'-n(n+2)y=0,
(21)

and b_n(x) the equation

x(x+4)y^('')+2(x+1)y^'-n(n+1)y=0.
(22)

These and several other identities involving derivatives and integrals of the polynomials are given by Swamy (1968).

See also

Brahmagupta Polynomial, Fibonacci Polynomial

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References

Lahr, J. "Fibonacci and Lucas Numbers and the Morgan-Voyce Polynomials in Ladder Networks and in Electric Line Theory." In Fibonacci Numbers and Their Applications (Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam). Dordrecht, Netherlands: Reidel, 1986.Morgan-Voyce, A. M. "Ladder Network Analysis Using Fibonacci Numbers." IRE Trans. Circuit Th. CT-6, 321-322, Sep. 1959.Swamy, M. N. S. "Properties of the Polynomials Defined by Morgan-Voyce." Fib. Quart. 4, 73-81, 1966a.Swamy, M. N. S. "More Fibonacci Identities." Fib. Quart. 4, 369-372, 1966b.Swamy, M. N. S. "Further Properties of Morgan-Voyce Polynomials." Fib. Quart. 6, 167-175, 1968.

Referenced on Wolfram|Alpha

Morgan-Voyce Polynomials

Cite this as:

Weisstein, Eric W. "Morgan-Voyce Polynomials." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Morgan-VoycePolynomials.html

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