Morgan-Voyce Polynomials

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

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) = [画像:sum_(k=0)^(n)(n+k+1; n-k)x^k]

(8)

b_n(x) = [画像:sum_(k=0)^(n)(n+k; n-k)x^k.]

(9)

Defining the matrix

[画像: Q=[x+2 -1; 1 0] ]

(10)

gives the identities

Q^n = [画像:[B_n -B_(n-1); B_(n-1) -B_(n-2)]]

(11)

Q^n-Q^(n-1) = [画像:[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) = [画像:(sin[(n+1)theta])/(sintheta)]

(15)

B_n(x) = [画像:(sinh[(n+1)phi])/(sinhphi)]

(16)

and

b_n(x) = [画像:(cos[1/2(2n+1)theta])/(cos(1/2theta))]

(17)

b_n(x) = [画像:(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) = [画像: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).

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

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 Resource. https://mathworld.wolfram.com/Morgan-VoycePolynomials.html

Morgan-Voyce Polynomials

See also

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Subject classifications