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ABSTRACT. The Fibonacci polynomials of order k are introduced and two expansions of them are obtained, in terms of the multlnomial and binomial coefficients, respectively.A relation between them and probability is also established. The present work generalizes results of [2] [4] and [5].

Upper and lower bounds are derived for the mode(s) ( , ) of the negative binomial distribution of order k, type I, with parameters r and p, say , ( , ), which are employed to establish an explicit formula for ( , ) in terms of r and k when p = 1/2. It is also shown as a direct consequence of the upper bound alone that ( , ) = when r = 1. The derivation of the bounds is based on a known recurrence relation satisfied by the probability mass function of , ( , ).