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.