| Popis: |
Let A, B, C denote rational numbers with AB ≠ 0 and m > n ≥ 3 arbitrary rational integers. We study the Diophantine equation APm(x) + Bpn(y) = C, in x, y ϵ Z , where {Pk(x)}I is one of the three classical continuous orthogonal polynomial families, i.e. Laguerre polynomials, Jacobi polynomials (including Gegenbauer, Legendre or Chebyshev polynomials) and Hermite polynomials. We prove that with exception of the Chebyshev polynomials for all such polynomial families there are at most finitely many solutions (x, y) ϵ Z 2 provided n > 4. The tools are besides the criterion [3], a theorem of Szeg— [14] on monotonicity of stationary points of polynomials which satisfy a second order Sturm-Liouville differential equation, (ax 2 + bx + c)y″ n (x) + (dx + e)y′(x) - λ n y n (x) = 0, n ϵ Z≥0. |
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