Generalized Powers of Substitution with Pre-Function Operators

An operator on formal power series of the form   S S    , where  is an invertible power series, and  is a series of the form is called a unipotent substitution with pre-function. Such operators, denoted by a pair   2 t t   ,    , 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   , a b   for every a b  .


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 S is a formal power series of the variable t.We recall that any formal power series of the form tS  is invertible with respect to the usual product of series.Its inverse is denoted by 1   and has the form for some In particular, the set of all series of the form 1 tT    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 t t T    0   , 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

 
,   where  is a unipotent series, and  is a unipotent substitution, and the operation   , , . The identity element is (1, t) .This group has been previously studied in [3][4][5], and is called the group of (unipotent) substitutions with pre-function.These substitutions with pre-function act on  for every series .In [3] is associated a doubly-infinite matrix

 
, S M   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 Actually in [3] no formal proof is given for the existence of such generalized powers for matrices or unipotent substitu-tions 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.

Differential and Delta Operators, and Their Associated Polynomial Sequences
By operator we mean a linear endomorphism of the -vector space of polynomials    2) for every non-constant polynomial .

   
be a differential operator.A normal family is said to be a basic family for if . 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 .
 is also a differential operator if, and only if, 0 0   and 1 0   .
Following [1], let us define a sequence of polynomials  be its unipotent part.Then we may consider generalized power (in particular, this explains the notation E  for the shift operator).We observe that for every integer , really coincides to the -th for     .We may also form where for every operat kind of generalized powers may be used to compute fractional power of the form or).This . The objective of this contributi pr of the existence on is to ovide a proof of such generalized powers for unipotent substitutions with pre-function.
Following [8], we may consider the following sub-set of An operator differential operators, called delta operators.A poly- For instance, the usual derivation  of poly s a delta operator.It can be proved th 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 D is a delta operator, then there exists a unique  -alg ra isomorphism from hat given a del erator , and a se  satisfies 0 0   and 1 0   .

Sheffer Sequences
riefly recall some definitions sequence of polynomials in In this section, we also b and results from [8].
Let be a delta operator and be its basic set.Let   where  

.
operator D with bas

Umbral Composition
This section is based on [11].Let . Th nt ope en ra tor, i a delta ope sequ given by

U tent nipo Sequences
The basic set   the (umbral) inverse of unipotent basic set is unipotent, and the (umbral) inverse of a a Sheffer seque 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 o so 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]).
on of the coefficient of on both sides, we obtain (since  is assumed to be a unipotent bstitution), and, by induction, (because there morphism between ), and Then, by identification of the coefficient of x

Gener Substi -Functio alized Powers of Unipotent tutions with Pre n
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 that substitution with pre-function.Moreover   q is rator potent basic set of the (unipotent) delta ope for every n .Therefore, here R id N   , w k is a lowering operator.Again for every    , we define In particular for   e basic set the unipotent delta o erator have also a degree at most Because the equations (1) 2) hold for every integ the polynomials (in the ble  , for every Moreover ,  ce  is a field of Remark 2. In particular, sin characteristic zero, for every q   , we may define fractional powers   , q   such as for instance   under multiplication, called the group of unipotent series.For a series of the form ,

1 
inverse of  is then denoted by   , it is possible to define a semi-direct product of groups by considering pairs


For instance, the usual derivation of polynomials is a differential operator.Moreover, let     , and let us define the shift-invariant operator E  as the unique linear map such that

1 
 (and, obviously, 0 0   ), i.e.,  is a unipotent substitution.A Sheffer sequence     .also clear from the previous sec-It is that nce f unipotent basic sets al tion (theorem 4)

Lemma 1 .
Let   ,   be a substitution with prefunction, and let    for each .As polynomials in the variable , their ost .As polynomials in th riable n .Thus for every k zero, and the above equations hold for sequence for every    .


is unip otent, and   is a unipotent substitution.It is also clear that whenever k    , then   means that for every ,
  we show