A Search for Solutions of a Functional Equation

Nash [4] used recursive sequences like the Fibonacci numbers, {Fn}, to investigate factors and divisibility. The Fibonacci numbers are defined by (1.1) $$ \begin{array}{*{20}{c}} {{F_n} = {F_{n - 1}} + {F_{n - 2}},}&{n > 2,} \end{array} $$ with F1 = F2 = 1. (The Lucas numbers, L n , satisfy the same linear homogeneous recurrence relation (1.1) but have initial conditions L 1, =1, L 2 = 3.) Brillhart, Montgomery and Silverman [1] also used the Fibonacci and Lucas numbers and the identity 1.2 $$ {F_{2n}} = {F_n}{L_n} $$ to investigate factorizations. Shannon, Loh and Horadam [8] generalized this in the context of the functional equation 1.3 $$ f(2k - {x^2}) = f(x)f( - x) $$ when k = 1. One of the authors (RPL) has attempted to find irreducible polynomial solutions over Q of degree n to the relation (1.3) and found that for k = 0, 1, nearly all solutions are proper divisors of recurrence relations. It is the purpose of this paper to draw some of the strands of this study together. In the next two sections we look at sequences which satisfy (1.3) when k = 0 and k = 1. Then we take some computer generated examples to consider aspects of the functions for general k.