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Date: 25-5-2019
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Date: 21-9-2018
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The Morgan-Voyce polynomials are polynomials related to the Brahmagupta and Fibonacci polynomials. They are defined by the recurrence relations
(1) |
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(2) |
for , with
(3) |
Alternative recurrences are
(4) |
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(5) |
with and , and
(6) |
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(7) |
The polynomials can be given explicitly by the sums
(8) |
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(9) |
Defining the matrix
(10) |
gives the identities
(11) |
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(12) |
Defining
(13) |
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(14) |
gives
(15) |
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(16) |
and
(17) |
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(18) |
The Morgan-Voyce polynomials are related to the Fibonacci polynomials by
(19) |
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(20) |
(Swamy 1968ab).
satisfies the ordinary differential equation
(21) |
and the equation
(22) |
These and several other identities involving derivatives and integrals of the polynomials are given by Swamy (1968).
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.
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