Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

It is well-known that Morgan-Voyce polynomials B _n(x) and b _n(x) satisfy both a Sturm-Liouville equation of second order and a three-term recurrence equation (SWAMY, M.: Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let A, B, C ∈ ? and {pk(x)} be a sequence of polynomials defined by

$\begin{gathered} p_0 (x) = 1 \hfill \\ p_1 (x) = x - c_0 \hfill \\ p_{n + 1} (x) = (x - c_n )p_n (x) - d_n p_{n - 1} (x), n = 1,2,..., \hfill \\ \end{gathered} $

with

$(c_0 ,c_n ,d_n ) \in \{ (A,A,B),(A + B,A,B^2 ),(A,Bn + A,\tfrac{1}{4}B^2 n^2 + Cn)\} $

with A ≠ 0, B > 0 in the first, B ≠ 0 in the second and C > ?¼B ² in the third case. We show that the Diophantine equation

Open image in new windowm > n ≥ 4,

Open image in new windowx, y.