1. Substitution of Formal Power Series
In this contribution we let 
denote any field of characteristic zero. We recall some basic definitions from [1,2]. The algebra of formal power series in the variable 
is denoted by
. In what follows we sometimes use the notation 
for 
to mean that 
is a formal power series of the variable t. We recall that any formal power series of the form 
for
and 
is invertible with respect to the usual product of series. Its inverse is denoted by 
and has the form 
for some
. In particular, the set of all series of the form 
forms a group under multiplication, called the group of unipotent series. For a series of the form
we may define for any other series 
an operation of substitution given by
. A unipotent substitution is a series of the form
. Such series form a group under the operation of substitution, called the group of unipotent substitutions (whenever
, a series 
is invertible under substitution, and the totality of such series forms a group under the operation of substitution called the group of substutions, and it is clear that the group of unipotent substitutions is a sub-group of this one). The inverse of 
is then denoted by 
and satisfies
. Finally, it is possible to define a semi-direct product of groups by considering pairs 
where 
is a unipotent series, and 
is a unipotent substitution, and the operation 
. The identity element is
. This group has been previously studied in [3-5], and is called the group of (unipotent) substitutions with pre-function. These substitutions with pre-function act on 
as follows: 
for every series
. In [3] is associated a doubly-infinite matrix 
to each such operator which defines a matrix representation of the group of substitutions with prefunction, and it is proved that there exists a oneparameter sub-group
. Therefore, it satisfies 
for every
, and
is the usual 
-th power of 
whenever
is an integer. It amounts that for every
, 
is the matrix representation of a substitution with prefunction say 
so that
. The authors of [3] then define
. Actually in [3] no formal proof is given for the existence of such generalized powers for matrices or unipotent substitutions with pre-function.
In this contribution, we provide a combinatorial proof for the existence of these generalized powers for unipotent substitutions with pre-function, and we show that this even forms a one-parameter sub-group. To achieve this objective we use some ingredients well-known in combinatorics such as delta operators, Sheffer sequences and umbral composition which are briefly presented in what follows (Sections 2, 3, 4 and 5). The Section 6 contains the proof of our result.
2. Differential and Delta Operators, and Their Associated Polynomial Sequences
By operator we mean a linear endomorphism of the 
-vector space of polynomials 
(in one indeterminate
). The composition of operators is denoted by a simple juxtaposition. If
, then we sometimes write 
to mean that 
is a polynomial in the variable
.
Let 
be a sequence of polynomials.
It is called a polynomial sequence if 
for every 
(in particular,
). It is clear that a polynomial sequence is thus a basis for
.
An operator 
is called a differential operator (see [6]) if 1) 
for every
.
2) 
for every non-constant polynomial
.
For instance, the usual derivation 
of polynomials is a differential operator. Moreover, let
, and let us define the shift-invariant operator 
as the unique linear map such that 
for every
. Then, 
is also a differential operator.
A polynomial sequence 
is said to be a normal family if 1)
.
2) 
for every
.
Let 
be a differential operator. A normal family 
is said to be a basic family for 
if
for every
. It is proved in [6] that for any differential operator admits is one and only one basic family, and, conversely, any normal family is the basic family of a unique differential operator. As an example, the normal family 
is the basic family of
.
Let 
be an operator such that for every non-zero polynomial
, 
(in particular, 
for every constant
). Such an operator is called a lowering operator (see [7]). For instance any differential operator is a lowering operator. Then given a lowering operator
, we may consider the algebra of formal power series 
of operators of the form 
where 
for every
.
The series 
converges to an operator of
in the topology of simple convergence (when 
has the discrete topology) since for every
, there exists 
such that for all
, 
, so that we may define
According to [6], if 
is a differential operator, then
if, and only if, 
commutes with
, i.e.,
. Moreover, if
, then
is also a differential operator if, and only if, 
and
.
Following [1], let us define a sequence of polynomials
by 
and 
for every integer
. For
, we denote by 
the value of the polynomial 
for
. Let 
be a lowering operator, and let 
be its unipotent part. Then we may consider generalized power
(in particular, this explains the notation 
for the shift operator). We observe that for every integer
, 
really coincides to the 
-th power 
of
. Moreover, 
for every
. We may also form
in such a way that for every
,
where for every 
with
, 
(it is a well-defined operator). This kind of generalized powers may be used to compute fractional power of the form 
for every
, 
(for instance,
). They satisfy the usual properties of powers:
,
. The objective of this contribution is to provide a proof of the existence of such generalized powers for unipotent substitutions with pre-function.
Following [8], we may consider the following sub-set of differential operators, called delta operators. A polynomial sequence 
is said to be of binomial-type if for every
,
An operator 
is a shift-invariant operator if for every
,
. Now, a delta operator 
is a shift invariant operator such that
. For instance, the usual derivation 
of polynomials is a delta operator. It can be proved that a delta operator is a differential operator. The basic family (uniquely) associated to a delta operator is called its basic set. Moreover, the basic set of a delta operator is of binomial-type, and to any polynomial sequence of binomial-type is uniquely associated a delta operator. If 
is a delta operator, then there exists a unique 
-algebra isomorphism from 
to the ring of shift-invariant operators 
that maps 
to
. In [8]
is proved that given a delta operator
, and a series 
with
, then 
is also a delta operator. Conversely, if 
is a shift-invariant operator (so that
), then if it is a delta operator, the unique series
such that 
satisfies 
and
.
3. Sheffer Sequences
In this section, we also briefly recall some definitions and results from [8].
Let 
be a sequence of polynomials in
. We define the exponential generating function of 
as
Let 
be a delta operator and 
be its basic set. Let 
with 
and 
such that
. Then from [8],
A polynomial sequence 
is said to be a Sheffer sequence (also called a polynomial sequence of type zero in [9] or a poweroid in [10]) if there exists a delta operator 
such that 1)
2) 
for every
.
Following [9], a polynomial sequence 
is a Sheffer sequence if, and only if, there exists a pair 
of formal power series in 
with 
invertible, and
, 
, such that
Remark 1. The basic set of a delta operator 
is a Sheffer sequence.
Let 
be a delta-operator with basic set
. Following [8], the following result holds.
Proposition 1. A polynomial sequence 
is a Sheffer sequence if, and only if, there exists an invertible shift-invariant operator 
such that 
for each
. Moreover, let 
be an invertible shift-invariant operator. Let 
be the unique formal power series such that
. Then, 
is invertible, and
where 
is the Sheffer sequence defined by
for each
, and 
is the unique formal power series such that
. Finally we also have the following characterization.
Proposition 2. Let 
be a polynomial sequence. It is a Sheffer sequence if, and only if, there exists a delta operator 
with basic set 
such that
4. Umbral Composition
This section is based on [11].
Let 
be a fixed polynomial sequence. Let us define an operator 
by 
for each
.
Since 
is a basis of
, this means that 
is a linear isomorphism of
. When 
is the basic set of a delta operator, then 
is referred to as an umbral operator, while if 
is a Sheffer sequence, then 
is said to be a Sheffer operator. An umbral operator maps basic sets to basic sets, while a Sheffer operator maps Sheffer sequences to Sheffer sequences.
Let 
be a polynomial sequence. For every
,
where 
is the coefficient of 
in the polynomial
. Let 
and 
be two polynomial sequences. Their umbral composition is defined as the polynomial sequence 
defined by
for each
. By simple computations, it may be proved that
. The set of all polynomial sequences becomes a (non-commutative) monoid under 
with 
as identity. We observe that if 
is the operator defined by 
for each
, then
. More generally, we have 
where 
is the
-th power of 
for the umbral composition (it is equal to a sequence say 
and we denote 
by
). Under umbral composition, the set of all Sheffer sequences is a (non-commutative) group, called the Sheffer group ([12]), and the set of all basic sequences is a sub-group of the Sheffer group.
From [8] we have the following result that combines delta operators, basis sets, Sheffer sequences and umbral composition.
Theorem 1. Let 
and 
be two delta operators with respective basic sets 
and
. Let 
and 
be two invertible shift-invariant operators. Let 
and 
be the Sheffer sequences defined by 
and 
for each
. Let 
be two invertible series such that
,
. Let 
be two formal power series with
, 
such that 
and
. Then,
is a shift-invariant operator, 
is a delta operator with basic sequence
. Finally, let 
be the Sheffer sequence given by
. Then, 
for each
.
It may be proved that if 
is the Sheffer sequence obtained from the delta operator 
with basic set 
and the invertible shift-invariant operator
, i.e., 
for each
, then the inverse
of 
with respect to the umbral composition is the basic set of the delta operator
, the inverse
of 
with respect to the umbral composition is the Sheffer sequence
.
5. Unipotent Sequences
The basic set 
of a delta operator 
is said to be unipotent if the unique series 
such that 
satisfies 
(and, obviously,
), i.e., 
is a unipotent substitution. A Sheffer sequence 
associated to a delta operator 
(with
,
) and an invertible shift-invariant operator 
(with 
invertible), i.e., 
for every 
where 
is the basic set of
, is said to be unipotent if 
is unipotent, and if 
is unipotent, i.e.,
. It is also clear from the previous section (theorem 4) that the (umbral) inverse of a unipotent basic set is unipotent, and the (umbral) inverse of a Sheffer sequence is also unipotent.
It is clear from theorem 4 that the group of basic sets under umbral composition is isomorphic to the group of substitutions. Moreover, the group of unipotent basic sets also is isomorphic to the group of unipotent substitutions. Likewise, the group of (unipotent) Sheffer sequences is isomorphic to the group of (unipotent) substitutions with pre-function (see also [12]).
Lemma 1. Let 
be a substitution with prefunction, and let 
be the Sheffer sequence and the basic set associated to the delta operator 
and the invertible shift-invariant operator 
(this means that 
is the basic set of
, and
for each
). Then, 
is a unipotent substitution with pre-function if, and only if, 
for every
.
Proof. Let us first assume that 
is a unipotent substitution with prefunction. We have 
for every basic set, so that
. Let
. We have
. Then, 
is equivalent to
.
By identification of the coefficient of 
on both sideswe obtain 
(since 
is assumed to be a unipotent substitution), and, by induction,
. Besides, we have
for each
. But 
(because there is a ring isomorphism between 
and
), and
, where
.
Then, by identification of the coefficient of
, we have
for every
. Conversely, let us assume that 
is the Sheffer sequence and the basic set associated to the delta operator 
and the invertible shift-invariant operator 
with
for every
. By construction we have 
so that
. Likewise, 
, so that
. □
6. Generalized Powers of Unipotent Substitutions with Pre-Function
The purpose of this section is to define 
for any 
and any unipotent substitution with pre-function
, and to prove that it is also a unipotent substitution with pre-function. Moreover we show that
is a one-parameter sub-group, i.e.,
for every
, and
.
Let 
be a unipotent substitution with prefunction of
. Let 
be the unipotent basic set of the (unipotent) delta operator
. Let 
be the unipotent Sheffer sequence associated to 
and the (unipotent) invertible shift-invariant operator
. Let 
be the umbral operator given by 
for all
, and let 
be the Sheffer operator defined by 
for all
. It is easily checked that for every integer
, 
and
. In particular, for each
, 
(by Lemma 5). Therefore, 
, where
for each
. The operator 
is actually a lowering operator. Then according to section 2it is possible to define 
for every
. Moreover, we have
.
For each
, let us define 
for every
. When
, we have 
. So that in this case, 
is the unipotent Sheffer sequence associated to
. This means that if
, and 
is the unipotent basic set of the (unipotent) delta operator
then 
for each
. Similarly, let
for every
. Therefore, 
, where 
is a lowering operator. Again for every
, we define
. For each
, we define 
for each
. In particular for
, 
, so that it is the basic set of the unipotent delta operator
. Clearly, 
for each
. Thus for every
, we have
(1)
(2)
Now, let 
be a variable commuting with 
and
, and let us define
and similarly,
for each
. As polynomials in the variable
, their degrees are at most
. As polynomials in the variable
, 
, 
,
and
have also a degree at most
. Because the equations (1) and (2) hold for every integer
, the polynomials (in the variable
)
and
are identically zero, and the above equations hold for every
. Therefore, 
is a polynomial sequence of binomial-type, and 
is a Sheffer sequence for every
. Moreover, for every
, we have
so that
. Similarly,
for every
. Moreover,
.
Therefore, 
and 
are one-parameter sub-groups. It follows that
and
(inverses under umbral operation).
We define 
as the pair of formal power series 
such that 
is the substitution that defines the delta operator 
with basic sequence
, and 
is the invertible series such that
for each
. Since 
and 
are unipotent sequences, it is clear that 
is unipotent, and 
is a unipotent substitution. It is also clear that whenever
, then
. Let us check that 
is a one-parameter subgroup of the group of unipotent substitutions with prefunction. This means that for every
,
First of all, by definition, 
is the unipotent substitution associated to the basic set
and therefore
. In a similar way, the series 
is uniquely associated to the Sheffer sequence
and to the basic set
. Again this means that
. Therefore, we obtain the expected result. It is also clear that
.
Remark 2. In particular, since 
is a field of characteristic zero, for every
, we may define fractional powers 
such as for instance
for each integer
.