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An operator on formal power series of the form *S* *μS* , where *μ* is an invertible power series, and σ is a series of the form t+（t^{2}） is called a unipotent substitution with pre-function. Such operators, denoted by a pair (*μ *，σ ） , form a group. The objective of this contribution is to show that it is possible to define a generalized powers for such operators, as for instance fractional powers σ for every .

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 formwe 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 [

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 [

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.

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 [

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 [

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 [

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 [

if, and only if, commutes with, i.e.,

. Moreover, if, then

is also a differential operator if, and only if, and.

Following [

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 [

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 [

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.

In this section, we also briefly recall some definitions and results from [

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 [

A polynomial sequence is said to be a Sheffer sequence (also called a polynomial sequence of type zero in [

Following [

Remark 1. The basic set of a delta operator is a Sheffer sequence.

Let be a delta-operator with basic set. Following [

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

This section is based on [

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 ([

From [

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.

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 [

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. □

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 operatorthen 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

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.