Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

Mathematica Slovaca

Robert Tichy¹  Thomas Stoll¹ 

关键词: Diophantine equation;    three-term recurrence;    orthogonal polynomial;    Morgan-Voyce polynomial;    Sturm-Liouville differential equation;   

DOI  :  10.2478/s12175-007-0051-2

学科分类：数学（综合）

来源: Slovenska Akademia Vied * Matematicky Ustav / Slovak Academy of Sciences, Mathematical Institute

【 摘 要 】

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 ∈ ℚ and {pk(x)} be a sequence of polynomials defined by $$egin{gathered} p_0 (x) = 1 hfill \ p_1 (x) = x - c_0 hfill \ p_{n + 1} (x) = (x - c_n )p_n (x) - d_n p_{n - 1} (x), n = 1,2,..., hfill \ end{gathered} $$ with $$(c_0 ,c_n ,d_n ) in { (A,A,B),(A + B,A,B^2 ),(A,Bn + A,frac{1}{4}B^2 n^2 + Cn)} $$ 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 with m > n ≥ 4, ≠ 0 has at most finitely many solutions in rational integers x, y.

【 授权许可】

Unknown