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Generalized Powers of Substitution with Pre-Function Operators

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Generalized Powers of Substitution with Pre-Function Operators Applied Mathematics, 2013, 4, 12-17 http://dx.doi.org/10.4236/am.2013.47A004 Published Online July 2013 (http://www.scirp.org/journal/am) Generalized Powers of Substitution with Pre-Function Operators Laurent Poinsot University Paris 13, Paris Sorbonne Cité, LIPN, CNRS, Villetaneuse, France Email: [email protected] Received April 13, 2013; revised May 13, 2013; accepted May 20, 2013 Copyright © 2013 Laurent Poinsot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT An operator on formal power series of the form SS , where is an invertible power series, and is a series of the form tt 2 is called a unipotent substitution with pre-function. Such operators, denoted by a pair , , form a group. The objective of this contribution is to show that it is possible to define a generalized powers for a a such operators, as for instance fractional powers , b for every . b Keywords: Formal Power Series; Formal Substitution; Riordan Group; Generalized Powers; Sheffer Sequences; Umbral Calculus 1. Substitution of Formal Power Series inverse of is then denoted by 1 and satisfies 11 In this contribution we let denote any field of t . Finally, it is possible to define a characteristic zero. We recall some basic definitions from semi-direct product of groups by considering pairs [1,2]. The algebra of formal power series in the variable , where is a unipotent series, and is a t is denoted by t . In what follows we sometimes unipotent substitution, and the operation 11, use the notation S t for S t to mean that S 22,, 1 2 212 . The identity element is is a formal power series of the variable t. We recall that (1, t) . This group has been previously studied in [3-5], any formal power series of the form tS for and is called the group of (unipotent) substitutions with and S t is invertible with respect to pre-function. These substitutions with pre-function act on the usual product of series. Its inverse is denoted by 1 t as follows: ,SS for every 1 series S . In [3] is associated a doubly-infinite matrix and has the form tV for some V t . In particular, the set of all series of the form 1t T M , to each such operator which defines a matrix forms a group under multiplication, called the group of representation of the group of substitutions with pre- unipotent series. For a series of the form tt2T , function, and it is proved that there exists a one- n parameter sub-group M , . Therefore, it sa- we may define for any other series S n t an n0 n tisfies MM,, M, for every , , and operation of substitution given by S n0n . A 2 unipotent substitution is a series of the form ttT . M , is the usual -th power of M , whenever Such series form a group under the operation of is an integer. It amounts that for every , M , is substitution, called the group of unipotent substitutions the matrix representation of a substitution with pre- (whenever 0 , a series tt2T is invertible function say , so that MM . The under substitution, and the totality of such series forms a , , group under the operation of substitution called the authors of [3] then define ,, . Actually group of substutions, and it is clear that the group of in [3] no formal proof is given for the existence of such unipotent substitutions is a sub-group of this one). The generalized powers for matrices or unipotent substitu- Copyright © 2013 SciRes. AM L. POINSOT 13 tions with pre-function. n the form n L where n for every n 0 . In this contribution, we provide a combinatorial proof n0 The series Ln converges to an operator of for the existence of these generalized powers for unipo- n0 n tent substitutions with pre-function, and we show that x in the topology of simple convergence (when this even forms a one-parameter sub-group. To achieve has the discrete topology) since for every p x , this objective we use some ingredients well-known in there exists np such that for all nn p , combinatorics such as delta operators, Sheffer sequences Lpn 0 , so that we may define and umbral composition which are briefly presented in np what follows (Sections 2, 3, 4 and 5). The Section 6 con- nn nnLp Lp . tains the proof of our result. nn00 According to [6], if D is a differential operator, then 2. Differential and Delta Operators, and if, and only if, commutes with D , i.e., Their Associated Polynomial Sequences D DD . Moreover, if DDn , then By operator we mean a linear endomorphism of the n0 n -vector space of polynomials x (in one indeter- is also a differential operator if, and only if, 0 0 minate x ). The composition of operators is denoted by a and 1 0 . simple juxtaposition. If p x , then we sometimes Following [1], let us define a sequence of polynomials write px to mean that p is a polynomial in the va- x x x 1 n by 0 and x i riable x . n 0 n 1 n 1!i0 n0 Let p x be a sequence of polynomials. n n0 for every integer n . For , we denote by It is called a polynomial sequence if deg pn n for n every n 0 (in particular, p0 ). It is clear that a polynomial sequence is thus a basis for x . x the value of the polynomial for x . Let L be An operator D is called a differential operator (see n [6]) if a lowering operator, and let UidL L be its 1) D 0 for every . unipotent part. Then we may consider generalized power 2) degDp deg p 1 for every non-constant poly- n nomial p . UL L (in particular, this explains n0 n For instance, the usual derivation of polynomials is a differential operator. Moreover, let , and let the notation E for the shift operator). We observe that k us define the shift-invariant operator E as the unique for every integer k , U really coincides to the k -th n n linear map such that E xx for every power UU of U . Moreover, U UU for n 0 . Then, E1 id is also a differential operator. k factors A polynomial sequence p is said to be a normal every , . We may also form n n0 family if n1 1 n 1) p0 1 . logUidL log LL n1 n 2) pn 0 0 for every n 0 . Let D be a differential operator. A normal family in such a way that for every , p is said to be a basic family for D if n n0 U exp logU Dpnn1 n1 p where for every LLn with 0 , for every n 0 . It is proved in [6] that for any differen- n0 n 0 tial operator admits is one and only one basic family, and, 1 exp n (it is a well-defined operator). This conversely, any normal family is the basic family of a n0 n! unique differential operator. As an example, the normal kind of generalized powers may be used to compute frac- n family x is the basic family of . a n0 Let L be an operator such that for every non-zero tional power of the form U b for every a , b polynomial p x , degLp deg p (in particu- 1 n n lar, L 0 for every constant ). Such an op- (for instance, UU ). They satisfy the usual proper- erator is called a lowering operator (see [7]). For in- ties of powers: Uid0 , UUU . The objective stance any differential operator is a lowering operator. of this contribution is to provide a proof of the existence Then given a lowering operator L , we may consider the of such generalized powers for unipotent substitutions algebra of formal power series L of operators of with pre-function. Copyright © 2013 SciRes. AM 14 L. POINSOT Following [8], we may consider the following sub-set , of formal power series in t with in- of differential operators, called delta operators. A poly- vertible, and 0 0 , 1 0 , such that nomial sequence p is said to be of binomial-type xt n n0 EGF s ;t t e . if for every n 0 , n n n Remark 1. The basic set of a delta operat or D is a pppnknkxy x y x,y. k =0 Sheffer sequence. Let D be a delta-operator with basic set p . An operator is a shift-invariant operator if for n n0 Following [8], the following result holds. every , E E . Now, a delta operator D Propos ition 1. A polynomial sequence s is a is a shift inva riant operator such that Dx . For in- n n0 Sheffer sequence if, and only if, there exists an invertible stance, the usual derivation of polynomials i s a delta 1 shift-invariant operator S such that snn Sp for each operator. It can be proved that a delta operator is a dif- n 0 . Moreover, let S be an invertible shift-invariant ferential operator. The basic family (uniquely) associated operator. Let t be the unique form al power to a delta operator is called its basic set. Moreover, the basic set of a delta operator is of binomial-type, and to series such that S . Then, is invertible, and any polynomial sequence of binomial-type is uniquely 1 1 EGF s ;t 1 ex associated a delta operator.
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