An inversion algorithm is derived for the dual Weierstrass-Laguerre transform
∫
0
∞
g
α
(
x
,
y
;
1
)
φ
(
y
)
y
α
e
−
y
/
(
α
+
1
)
d
y
\int _0^\infty {{g_\alpha }(x,y;1)\varphi (y){y^\alpha }{e^{ - y}}/(\alpha + 1)dy}
, where the function
g
α
(
x
,
y
,
t
)
{g_\alpha }(x,y,t)
is associated with the source solution of the Laguerre differential heat equation
x
u
x
x
(
x
,
t
)
=
(
α
+
1
−
x
)
u
x
(
x
,
t
)
=
u
t
(
x
,
t
)
x{u_{xx}}(x,t) = (\alpha + 1 - x){u_x}(x,t) = {u_t}(x,t)
. Correspondingly, sufficient conditions are established for a function to be represented by a Weierstrass-Laguerre Stieltjes transform
∫
0
∞
g
α
(
x
,
y
;
1
)
d
β
(
y
)
\int _0^\infty {{g_\alpha }(x,y;1)\;d\beta (y)}
of a nondecreasing function
β
\beta
.