A polynomial in a single variable is uniquely determined by its derivatives of even order at 0 and 1. More precisely, such an univariate polynomial can be written and a finite sum of
f
(
2
n
)
(
0
)
Λ
n
(
1
−
z
)
f^{(2n)}(0) \Lambda _n(1-z)
and
f
(
2
n
)
(
1
)
Λ
n
(
z
)
f^{(2n)}(1) \Lambda _n(z)
, (
n
≥
0
n\ge 0
), where the
Λ
n
(
z
)
\Lambda _n(z)
are the Lidstone polynomials defined by the conditions
(
d
d
z
)
2
k
Λ
n
(
0
)
=
0
and
(
d
d
z
)
2
k
Λ
n
(
1
)
=
δ
k
,
n
,
k
≥
0
,
n
≥
0.
\begin{equation*} \left (\frac {\mathrm {d}}{\mathrm {d}z}\right )^{2k} \Lambda _n(0)=0\text { and } \left (\frac {\mathrm {d}}{\mathrm {d}z}\right )^{2k} \Lambda _n(1)=\delta _{k,n},\quad k\ge 0, \; n\ge 0. \end{equation*}
We generalize this theory to
n
n
variables, replacing the two points
0
0
,
1
1
in
C
\mathbb {C}
with
n
+
1
n+1
points
e
_
0
,
e
_
1
,
…
,
e
_
n
{\underline {e}}_0,{\underline {e}}_1,\dots ,{\underline {e}}_n
in
C
n
\mathbb {C}^n
, where
e
_
0
{\underline {e}}_0
is the origin of
C
n
\mathbb {C}^n
and
e
_
1
,
…
,
e
_
n
{\underline {e}}_1,\dots ,{\underline {e}}_n
the canonical basis of
C
n
\mathbb {C}^n
. By selecting a suitable subset of even order derivatives at these
n
+
1
n+1
points, we show that any polynomial in
n
n
variables has a unique expansion. We obtain generating series for these sequences of polynomials and we deduce an expansion for entire functions in
C
n
\mathbb {C}^n
of exponential type
>
π
>\pi
. We extend to several variables results due to Lidstone [Proceedings Edinburgh Math. Soc., II. Ser. (2)2, 16-19 (1930)], Poritsky [Trans. Amer. Math. Soc. 34 (1932), pp. 274–331], Whittaker [Proc. London Math. Soc. (2) 36 (1934), pp. 451–469], Schoenberg [Bull. Amer. Math. Soc. 42 (1936), pp. 284–288], Buck [Proc. Amer. Math. Soc. 6 (1955), pp. 793–796]. We also show that our results are, to a certain extent, best possible.