Morgan-Voyce Polynomials

👁 DOWNLOAD Mathematica Notebook
Download Wolfram Notebook

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).

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

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