Fibonacci polynomials

The polynomials (cf. [a1] and [a4]) given by

(a1)

They reduce to the Fibonacci numbers for and they satisfy several identities, which may be easily proved by induction, e.g.:

(a2)

(a3)

(a4)

(a5)

where

so that ; and

(a6)

where denotes the greatest integer in .

W.A. Webb and E.A. Parberry [a14] showed that the are irreducible polynomials over the ring of integers if and only if is a prime number (cf. also Irreducible polynomial). They also found that , , are the roots of (see also [a2]). M. Bicknell [a1] proved that divides if and only if divides . V.E. Hoggatt Jr., and C.T. Long [a3] introduced the bivariate Fibonacci polynomials by the recursion

(a7)

and they showed that the are irreducible over the rational numbers if and only if is a prime number. They also generalized (a5) and proved that

(a8)

In a series of papers, A.N. Philippou and his associates (cf. [a5], [a6], [a7], [a8], [a9], [a10], [a11], [a12], [a13]) introduced and studied Fibonacci, Fibonacci-type and multivariate Fibonacci polynomials of order , and related them to probability and reliability. Let be a fixed positive integer greater than or equal to . The Fibonacci polynomials of order , , are defined by

(a9)

For these reduce to , and for these reduce to , the Fibonacci numbers of order (cf. [a11]). Deriving and expanding the generating function of , they [a12] obtained the following generalization of (a6) in terms of the multinomial coefficients (cf. Multinomial coefficient):

(a10)

where the sum is taken over all non-negative integers such that . They also obtained a simpler formula in terms of binomial coefficients. As a byproduct of (a10), they were able to relate these polynomials to the number of trials until the occurrence of the th consecutive success in independent trials with success probability . For this formula reduces to

(a11)

The Fibonacci-type polynomials of order , , defined by

(a12)

have simpler multinomial and binomial expansions than . The two families of polynomials are related by

(a13)

Furthermore, with ,

(a14)

Assuming that the components of a consecutive -out-of-: -system are ordered linearly and function independently with probability , Philippou [a6] found that the reliability of the system, , is given by

(a15)

If the components of the system are ordered circularly, then its reliability, , is given by (cf. [a10])

(a16)

Next, denote by the number of independent trials with success probability until the occurrence of the th th consecutive success. It is well-known [a5] that has the negative binomial distribution of order with parameters and . Philippou and C. Georghiou [a9] have related this probability distribution to the -fold convolution of with itself, say , as follows:

(a17)

which reduces to (a14) for , and they utilized effectively relation (a17) for deriving two useful expressions, a binomial and a recurrence one, for calculating the above probabilities.

Let . The multivariate Fibonacci polynomials of order (cf. [a8]), , are defined by the recurrence

(a18)

For , , , and for , . These polynomials have the following multinomial expansion:

(a19)

where the sum is taken over all non-negative integers such that . Let the random variable be distributed as a multi-parameter negative binomial distribution of order (cf. [a7]) with parameters (, for and ). Philippou and D.L. Antzoulakos [a8] showed that the -fold convolution, , of with itself is related to this distribution by

(a20)

Furthermore, they have effectively utilized relation (a20) in deriving a recurrence for calculating the above probabilities.

References

[a1]	M. Bicknell, "A primer for the Fibonacci numbers VII" Fibonacci Quart. , 8 (1970) pp. 407–420
[a2]	V.E. Hoggatt Jr., M. Bicknell, "Roots of Fibonacci polynomials" Fibonacci Quart. , 11 (1973) pp. 271–274
[a3]	V.E. Hoggatt Jr., C.T. Long, "Divisibility properties of generalized Fibonacci polynomials" Fibonacci Quart. , 12 (1974) pp. 113–120
[a4]	E. Lucas, "Theorie de fonctions numeriques simplement periodiques" Amer. J. Math. , 1 (1878) pp. 184–240; 289–321
[a5]	A.N. Philippou, "The negative binomial distribution of order and some of its properties" Biom. J. , 26 (1984) pp. 789–794
[a6]	A.N. Philippou, "Distributions and Fibonacci polynomials of order , longest runs, and reliability of concecutive--out-of-: systems" A.N. Philippou (ed.) G.E. Bergum (ed.) A.F. Horadam (ed.) , Fibonacci Numbers and Their Applications , Reidel (1986) pp. 203–227
[a7]	A.N. Philippou, "On multiparameter distributions of order " Ann. Inst. Statist. Math. , 40 (1988) pp. 467–475
[a8]	A.N. Philippou, D.L. Antzoulakos, "Multivariate Fibonacci polynomials of order and the multiparameter negative binomial distribution of the same order" G.E. Bergum (ed.) A.N. Philippou (ed.) A.F. Horadam (ed.) , Applications of Fibonacci Numbers , 3 , Kluwer Acad. Publ. (1990) pp. 273–279
[a9]	A.N. Philippou, C. Georghiou, "Convolutions of Fibonacci-type polynomials of order and the negative binomial distributions of the same order" Fibonacci Quart. , 27 (1989) pp. 209–216
[a10]	A.N. Philippou, F.S. Makri, "Longest circular runs with an application in reliability via the Fibonacci-type polynomials of order k" G.E. Bergum (ed.) A.N. Philippou (ed.) A.F. Horadam (ed.) , Applications of Fibonacci Numbers , 3 , Kluwer Acad. Publ. (1990) pp. 281–286
[a11]	A.N. Philippou, A.A. Muwafi, "Waiting for the kth consecutive success and the Fibonacci sequence of order " Fibonacci Quart. , 20 (1982) pp. 28–32
[a12]	A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci polynomials of order , multinomial expansions and probability" Internat. J. Math. Math. Sci. , 6 (1983) pp. 545–550
[a13]	A.N. Philippou, C. Georghiou, G.N. Philippou, "Fibonacci-type polynomials of order with probability applications" Fibonacci Quart. , 23 (1985) pp. 100–105
[a14]	W.A. Webb, E.A. Parberry, "Divisibility properties of Fibonacci polynomials" Fibonacci Quart. , 7 (1969) pp. 457–463