Generalized Bivariate Fibonacci Polynomials

We define generalized bivariate polynomials, from which specifying initial conditions the bivariate Fibonacci and Lucas polynomials are obtained. Using essentially a matrix approach we derive identities and inequalities that in most cases generalize known results. 1 Antefacts The generalized bivariate Fibonacci polynomial may be defined as Hn(x, y) = xHn−1(x, y) + yHn−2(x, y), H0(x, y) = a0, H1(x, y) = a1. We assume x, y 6= 0, as well as x2 + 4y 6= 0. If we set a0 = 0, a1 = 1 we obtain the bivariate Fibonacci polynomials Fn(x, y); with a0 = 2, a1 = x we get the bivariate Lucas polynomials. Ln(x, y). The univariate polynomials are obtained setting y = 1. The characteristic polynomial is f(t) = t − tx− y, and the roots are (see for example [3]) α = α(x, y) = x+ √ x2 + 4y 2 , β = β(x, y) = x− √ x2 + 4y 2 . Note that α + β = x, αβ = −y. Furthermore α > β; if y > 0 then α > 0 and β < 0 and if x > 0 then α > |β|, while if x < 0 then α < |β|; if y < 0 then α and β have the same sign: if x > 0 they are both positive, while if x < 0 they are both negative. 1 The ordinary generating function g(t) is g(t) = a0 + (a1 − a0x)t 1− xt− yt2 , which can be obtained through the usual trick: write g(t) = H0 +H1t+H2t 2 + · · · then obtain xtg(t) and yt2g(t), subtract them from g(t), use the recurrence to get rid of all summands except the first two and impose the initial conditions. Using the Rational Expansion Theorem (see [5]) we get the Binet’s form Hn(x, y) = (a1 − βa0) α α− β − (a1 − αa0) β α− β . (1) Hence Fn(x, y) = α α− β − β α− β , Ln(x, y) = α n + β. In the sequel we will write simplyHn, Fn and Ln instead ofHn(x, y), Fn(x, y) and Ln(x, y). The sequence can be extended to negative subscripts by defining H−n = − x y H−(n−1) + 1 y H−(n−2). In this case the characteristic polynomial is the reflected polynomial (see [5]) of f(t): hence the roots are 1 α and 1 β . Define the matrix A = [