Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials

Rok, strany: 2008, 11 - 18

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.: \textit{Further properties of Morgan-Voyce polynomials}, Fibonacci Quart. \textbf{6} (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let $A, B, C\in \mathbb{Q}$ and $\{p_k(x)\}$ be a sequence of polynomials defined by \vspace{-2mm} \begin{align*} p₀(x)&=1\\[-1mm] p₁(x)&=x-c₀\\[-1mm] p_n+1(x)&=(x-c_n)p_n(x)-d_n p_n-1(x),   n=1,2,…, \end{align*} \vspace{-2mm} \noindent with

$$ (c₀,c_n,d_n)\in \{(A,A,B),(A+B,A,B²), (A,Bn+A,\tfrac{1}{4} B² n²+C n)\} $$

with $A\neq 0$, $B>0$ in the first, $B\neq 0$ in the second and $C>-((1) / (4)) B²$ in the third case. We show that the Diophantine equation

$$ \mathcal{A}p_m(x)+\mathcal{B}p_n(y)=\mathcal{C} $$

with $m>n≥ 4$, $\mathcal{A}, \mathcal{B}, \mathcal{C}\in \mathbb{Q}$, $\mathcal{A}\mathcal{B}\neq 0$ has at most finitely many solutions in rational integers $x,y$.

ISO 690:

Stoll, T., Tichy, R. 2008. Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials. In Mathematica Slovaca, vol. 58, no.1, pp. 11-18. 0139-9918.

APA:

Stoll, T., Tichy, R. (2008). Diophantine equations for Morgan-Voyce and other modified orthogonal polynomials. Mathematica Slovaca, 58(1), 11-18. 0139-9918.