Operator
is a polynomial in
but
so
can also be expressed as a polynomial in terms of shift operators
,
,
, and
converting the differential equation for
into a difference equation among contiguous instances of
which we call a contiguity relation.
Let
stand for
,
,
, or
and
stand for
,
,
, or
respectively. If we express
as a polynomial in
, then we get
where the
signs depend on
,
,
,
and whether
is
,
,
,
and
.
These results let us define
The coefficients of these polynomials in
,
,
,
and
are defined when
Operators
are defined for all
,
,
, and
.
