The Poisson-Laguerre transform of a function
ϕ
\phi
is given by
\[
u
(
n
,
t
)
=
∑
m
=
0
∞
g
(
n
,
m
;
t
)
ϕ
(
m
)
m
!
Γ
(
m
+
α
+
1
)
u(n,t) = \sum \limits _{m = 0}^\infty {g(n,m;t)\phi (m)\frac {{m!}}{{\Gamma (m + \alpha + 1)}}}
\]
where
g
g
, defined by
\[
g
(
n
,
m
;
t
)
=
Γ
(
n
+
m
+
α
+
1
)
n
!
m
!
t
m
+
m
(
1
+
t
)
n
+
m
+
α
+
1
⋅
2
F
1
(
−
n
,
−
m
;
−
n
−
m
−
α
;
1
−
1
t
2
)
,
g(n,m;t) = \frac {{\Gamma (n + m + \alpha + 1)}}{{n!m!}}\frac {{{t^{m + m}}}}{{{{(1 + t)}^{n + m + \alpha + 1}}}}{ \cdot _2}{F_1}\left ( { - n, - m; - n - m - \alpha ;1 - \frac {1}{{{t^2}}}} \right ),
\]
s the associated function of the source solution
g
(
n
;
t
)
=
g
(
n
,
0
;
t
)
g(n;t) = g(n,0;t)
of the Laguerre difference heat equation
\[
∇
n
u
(
n
,
t
)
=
u
t
(
n
,
t
)
,
{\nabla _n}u(n,t) = {u_t}(n,t),
\]
with
\[
∇
n
f
(
n
)
=
(
n
+
1
)
f
(
n
+
1
)
=
(
2
n
+
α
+
1
)
f
(
n
)
+
(
n
+
α
)
f
(
n
−
1
)
.
{\nabla _n}f(n) = (n + 1)f(n + 1) = (2n + \alpha + 1)f(n) + (n + \alpha )f(n - 1).
\]
A simple algorithm for the inversion of the transform
(
∗
)
(*)
is derived. For
m
=
0
m = 0
, the transform
(
∗
)
(*)
is basically a power series so that the inversion algorithm leads to a useful identity involving binomial coefficients. In addition, a subclass of functions is characterized that is representable by a Poisson-Laguerre transform
(
∗
)
(*)
.