We consider the generators
L
k
L_k
of Heckman-Opdam diffusion processes in the compact and non-compact case in
N
N
dimensions for root systems of type
A
A
and
B
B
, with a multiplicity function of the form
k
=
κ
k
0
k=\kappa k_0
with some fixed value
k
0
k_0
and a varying constant
κ
∈
[
0
,
∞
[
\kappa \in \,[0,\infty [
. Using elementary symmetric functions, we present polynomials which are simultaneous eigenfunctions of the
L
k
L_k
for all
κ
∈
]
0
,
∞
[
\kappa \in \,]0,\infty [
. This leads to martingales associated with the Heckman-Opdam diffusions
(
X
t
,
1
,
…
,
X
t
,
N
)
t
≥
0
(X_{t,1},\ldots ,X_{t,N})_{t\ge 0}
. As our results extend to the freezing case
κ
=
∞
\kappa =\infty
with a deterministic limit after some renormalization, we find formulas for the expectations
E
(
∏
j
=
1
N
(
y
−
X
t
,
j
)
)
,
\mathbb E(\prod _{j=1}^N(y-X_{t,j})),
y
∈
C
y\in \mathbb C
.