Polynomials whose coefficients are generalized Leonardo numbers

Abstract

Let an be defined recursively by an = an–1 + an–2 + pn + q if n ≥ 2, with a0 = a1 = 1, where p and q are constants. The p = 0 case of an corresponds to what have been termed the generalized Leonardo numbers, the q = 1 case of which are the classical Leonardo numbers. Let Ln(x)=∑i=0naixn−i, which we refer to as a generalized Leonardo coefficient polynomial. Here, it is shown, under some mild assumptions on p and q, that Ln(x) has exactly one real zero if n is odd and no real zeros if n is even. This yields a previous result for the Fibonacci coefficient polynomials when p = q = 0 and a new analogous result for the Leonardo coefficient polynomials when p = 0, q = 1. Further, when p = 0 and –1 < q ≤ 1 + ρ, we show that the sequence consisting of the real zeros of the Ln(x) for n odd is strictly decreasing with limit –ρ, where ρ=1+52. Finally, under the same assumptions on p and q, we are able to show the convergence in modulus of all the zeros of Ln(x).