Volume 12, 2006, Number 1

This paper looks at some of the properties of the auxiliary equation associated with the plastic number which, in turn, is related to the sequences of numbers {P_n}, {Q_n} and {R_n}, respectively, defined by
P_n = P_{n − 2} + P_{n − 3}, n > 3, P₁ = 1, P₂ = 1, P₃ = 1,
Q_n = Q_{n − 2} + Q_{n − 3}, n > 3, Q₁ = 0, Q₂ = 2, Q₃ = 3,
R_n = R_{n − 2} + R_{n − 3}, n > 3, R₁ = 1, R₂ = 0, R₃ = 1.
The dominant root of the associated auxiliary equation is found by a contraction process related to Bernoulli’s iteration and the Jacobi–Perron Algorithm. The latter is one way of generalizing the ordinary continued fraction algorithm and an alternative way is explored which also relates to the auxiliary equations of the sequences. Various methods for reduction of the order of the cubic auxiliary equation are also considered.

This paper considers some q-extensions of binomial coefficients. Some of the results are applied to some generalized Fibonacci numbers, and others are included as ideas for further investigation, particularly, particularly into q-Bernoulli polynomials.

For x > 0 one define the function S(x) = min{m ∈ ℕ | x ≤ m!}. We prove that for x > √13! the interval (S(x), S(x²)) contains at least a prime number and that for real x, y > 0 the inequality S(x) + S(y) ≥ S(xy) holds true. We also study the convergence of a couple of number series involving S(x).