Piero Filipponi

INCOMPLETE FIBONACCI AND LUCAS NUMBERS

A particular use of well-known combinatorial expressions for Fibonacci and Lucas numbers gives rise to two interesting classes of integers (namely, the numbersFn(k) andLn(k)) governed by the integral parametersn andk. After establishing the main properties of these numbers and their interrelationship, we study some congruence properties ofLn(k), one of which leads to a supposedly new characterisation of prime numbers. A glimpse of possible generalisations and further avenues of research is also caught.

Derivative Sequences of Fibonacci and Lucas Polynomials

Let us consider the Fibonacci polynomials U n(x) and the Lucas polynomials V n (x) (or simply U n and Vn, if there is no danger of confusion) defined as

On certain Fibonacci‐type sums

{U_n} = x{U_{n - 1}} + {U_{n - 2}}({U_0} = 0,{U_1} = 1)

A note on the improper use of a formula for Fibonacci numbers

(1.1) and

Partial Derivative Sequences of Second-Order Recurrence Polynomials

{V_n} = x{V_{n - 2}}({V_0} = 2,V = x)

An algorithm for computing functions of triangular matrices

(1.2) where x is an indeterminate. These polynomials are a natural extension of the numbers U n(m) and V n(m) considered in [1]. They have already been considered elsewhere (e.g., see [6]).