We look for differential equations satisfied by the generalized Jacobi polynomials
{
P
n
α
,
β
,
M
,
N
(
x
)
}
n
=
0
∞
\{ P_n^{\alpha ,\beta ,M,N}(x)\} _{n = 0}^\infty
which are orthogonal on the interval
[
−
1
,
1
]
[- 1,1]
with respect to the weight function
\[
Γ
(
α
+
β
+
2
)
2
α
+
β
+
1
Γ
(
α
+
1
)
Γ
(
β
+
1
)
(
1
−
x
)
α
(
1
+
x
)
β
+
M
δ
(
x
+
1
)
+
N
δ
(
x
−
1
)
,
\frac {{\Gamma (\alpha + \beta + 2)}}{{{2^{\alpha + \beta + 1}}\Gamma (\alpha + 1)\Gamma (\beta + 1)}}{(1 - x)^\alpha }{(1 + x)^\beta } + M\delta (x + 1) + N\delta (x - 1),
\]
where
α
>
−
1
\alpha > - 1
,
β
>
−
1
\beta > - 1
,
M
≥
0
M \geq 0
, and
N
≥
0
N \geq 0
. In the special case that
β
=
α
\beta = \alpha
and
N
=
M
N = M
we find all differential equations of the form
\[
∑
i
=
0
∞
c
i
(
x
)
y
(
i
)
(
x
)
=
0
,
y
(
x
)
=
P
n
α
,
α
,
M
,
M
(
x
)
,
\sum \limits _{i = 0}^\infty {{c_i}(x){y^{(i)}}(x) = 0,\quad y(x) = P_n^{\alpha ,\alpha ,M,M}(x),}
\]
where the coefficients
{
c
i
(
x
)
}
i
=
1
∞
\{ {c_i}(x)\} _{i = 1}^\infty
are independent of the degree n. We show that if
M
>
0
M > 0
only for nonnegative integer values of
α
\alpha
there exists exactly one differential equation which is of finite order
2
α
+
4
2\alpha + 4
. By using quadratic transformations we also obtain differential equations for the polynomials
{
P
n
α
,
±
1
/
2
,
0
,
N
(
x
)
}
n
=
0
∞
\{ P_n^{\alpha , \pm 1/2,0,N}(x)\} _{n = 0}^\infty
for all
α
>
−
1
\alpha > - 1
and
N
≥
0
N \geq 0
.