Let
The differential equation for
becomes
Now
is a polynomial in
but
so
can also be expressed as a polynomial in terms of shift operators
and
converting the differential equation
into a difference equation among contiguous instances of
which we call a contiguity relation. Operators
are defined if
and
respectively.
If we express
as a polynomial in
, then we get
which has degree .
If we express
as a polynomial in
, we get
which has degree at most .
These results let us define
The coefficients of these polynomials in
and
are defined when
Operators
and
for
are defined if
and
respectively. Hence,
is defined if
and
is distinct from all
.
is defined if
and
is distinct from all
. Recall that
.
