Harmonic divisors and rationality of zeros of Jacobi polynomials

Let Pn (α,β ) ( x ) be the Jacobi polynomial of degree n with parameters αβ The main result of the paper states the following: If b≠ 1 ; 3 and c are non-zero rel- atively prime natural numbers then P ( k +( d 3) = 2 ;k +( d 3) = 2) n p b=c 6 ≠ 0 for all natural numbers d;n and k 2 N 0 : Moreover, under the above assumption, the polynomial Q ( x ) = b c x 2 1 + ::: + x 2 d 1 + b c 1 x 2 d is not a harmonic divisor, and the Dirichlet problem for the cone f Q ( x ) < 0 g has polynomial harmonic solutions for polynomial data functions.AD 26/03/201