Abstract
In this paper we consider the differential difference operator
Y
α
,
β
=
-
z
d
d
z
+
2
(
α
+
β
+
1
)
+
(
α
-
β
)
z
1
-
z
2
-
(
α
+
β
+
1
)
1
-
τ
2
,
where (
τf
)[
z
] =
f
[
z
-1
] The eigenfunction of this operator equal to 1 at 1 is called nonsymmetric Jacobi function. We define the finite continuous nonsymmetric Jacobi transform as an extension of the nonsymmetric Fourier Jacobi series. The basic properties including the inversion formula for this transform are studied. We also derive a sampling expansion associated to
Y
α, β
.
2010 Mathematics Subject Classification: 33C45, 3352, 94A20
.