Lanczos and Ortiz placed the canonical polynomials (c.p.’s) in a central position in the Tau Method. In addition, Ortiz devised a recursive process for determining c.p.’s consisting of a generating formula and a complementary algorithm coupled to the formula. In this paper a) We extend the theory so as to include in the formalism also the ordinary linear differential operators with polynomial coefficients
D
D
with negative height
h
=
max
n
∈
N
{
m
n
−
n
}
>
0
,
\begin{equation*}h=\underset {{n\in N}}{\max } \{m_{n}-n\}>0, \end{equation*}
where
m
n
m_{n}
denotes the degree of
D
x
n
Dx^{n}
. b) We establish a basic classification of the c.p.’s
Q
m
(
x
)
Q_{m}(x)
and their orders
m
∈
M
m\in M
, as primary or derived, depending, respectively, on whether
∃
n
∈
N
:
m
n
=
m
\exists n\in \mathbf {N}\colon m_{n}=m
or such
n
n
does not exist; and we state a classification of the indices
n
∈
N
n\in \mathbf {N}
, as generic
(
m
n
=
n
+
h
)
(m_{n}=n+h)
, singular
(
m
n
>
n
+
h
)
(m_{n}>n+h)
, and indefinite
(
D
x
n
≡
0
)
(Dx^{n}\equiv 0)
. Then a formula which gives the set of primary orders is proved. c) In the rather frequent case in which all c.p.’s are primary, we establish, for differential operators
D
D
with any height
h
h
, a recurrency formula which generates bases of the polynomial space and their multiple c.p.’s arising from distinct
x
n
x^{n}
,
n
∈
N
n\in N
, so that no complementary algorithmic construction is needed; the (primary) c.p.’s so produced are classified as generic or singular, depending on the index
n
n
. d) We establish the general properties of the multiplicity relations of the primary c.p.’s and of their associated indices. It becomes clear that Ortiz’s formula generates, for
h
≥
0
h\ge 0
, the generic c.p.’s in terms of the singular and derived c.p.’s, while singular and derived c.p.’s and the multiples of distinct indices are constructed by the algorithm.