Morgan-Voyce Polynomial

Polynomials related to the Brahmagupta 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 $n\geq 1$, with

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

Alternative recurrences are

$$B_{n+1}B_{n-1}-{B_n}^2=-1$$ (4)

$$b_{n+1}b_{n-1}-{b_n}^2=x.$$ (5)

The polynomials can be given explicitly by the sums

$$B_n(x) = \sum_{k=0}^n {n+k-1\choose n-k}x^k$$ (6)

$$b_n(x) = \sum_{k=0}^n {n+k\choose n-k}x^k.$$ (7)

Defining the Matrix

$${\hbox{\sf Q}}=\left[{\matrix{x+2 & -1\cr 1 & 0\cr}}\right]$$ (8)

gives the identities

$${\hbox{\sf Q}}^n=\left[{\matrix{B_n & -B_{n-1}\cr B_{n-1} & -B_{n-2}\cr}}\right]$$ (9)

$${\hbox{\sf Q}}^n-{\hbox{\sf Q}}^{n-1}=\left[{\matrix{b_n & -b_{n-1}\cr b_{n-1} & -b_{n-2}\cr}}\right].$$ (10)

Defining

$$\cos\theta = {\textstyle{1\over 2}}(x+2)$$ (11)

$$\cosh\phi = {\textstyle{1\over 2}}(x+2)$$ (12)

gives

$$B_n(x) = {\sin[(n+1)\theta]\over\sin\theta}$$ (13)

$$B_n(x) = {\sinh[(n+1)\phi]\over\sinh\phi}$$ (14)

and

$$b_n(x) = {\cos[{\textstyle{1\over 2}}(2n+1)\theta]\over\cos({\textstyle{1\over 2}}\theta)}$$ (15)

$$b_n(x) = {\cosh[{\textstyle{1\over 2}}(2n+1)\phi]\over\cosh({\textstyle{1\over 2}}\theta)}.$$ (16)

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

$$b_n(x^2) = F_{2n+1}(x)$$ (17)

$$B_n(x^2) = {1\over x}F_{2n+2}(x)$$ (18)

(Swamy 1968).

$B_n(x)$ satisfies the Ordinary Differential Equation

$$x(x+4)y''+3(x+2)y'-n(n+2)y=0,$$ (19)

and $b_n(x)$ the equation

$$x(x+4)y''+2(x+1)y'-n(n+1)y=0.$$ (20)

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

See also Brahmagupta Polynomial, Fibonacci Polynomial

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, 1966.

Swamy, M. N. S. ``More Fibonacci Identities.'' Fib. Quart. 4, 369-372, 1966.

Swamy, M. N. S. ``Further Properties of Morgan-Voyce Polynomials.'' Fib. Quart. 6, 167-175, 1968.