Generalized Powers of Substitution with Pre-Function Operators

Applied Mathematics, 2013, 4, 12-17 doi:10.4236/am.2013.47A004 Published Online July 2013 (http://www.scirp.org/journal/am)

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  n0    n tisfies MMM,,,  for every ,    , and operation of substitution given by S    n0n . 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 n0 The series  Ln converges to an operator of for the existence of these generalized powers for unipo- n0 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. nn00 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 n0 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 px 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!i0 n0   Let p  x  be a sequence of polynomials. n n0  for every integer n . For    , we denote by It is called a polynomial sequence if deg pnn  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) degDp deg  p 1 for every non-constant poly-   n nomial p . ULL   (in particular, this explains n0 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, UUU for n  0 . Then, Eid1  is also a differential operator. k factors A polynomial sequence p is said to be a normal every ,    . We may also form n n0 family if n1 1 n 1) p0  1 . logUidL log   LL   n1 n   2) pn 00 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 n0 UU  exp log Dpnn1  n1 p where for every LLn    with   0 , for every n  0 . It is proved in [6] that for any differen- n0 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 n0 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 n0 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 , degLp 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.

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 n0 EGF s  ;t   t e . if for every n  0 ,  n n  n Remark 1. The basic set of a delta operator D is a pppnknkxy  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 n0 Following [8], the following result holds. every    , EE  . Now, a delta operator D  Proposition 1. A polynomial sequence s is a is a shift invariant operator such that Dx  . For in- n n0  Sheffer sequence if, and only if, there exists an invertible stance, the usual derivation  of polynomials is 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 formal 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. If D is a delta operator, n n   then there exists a unique  -algebra isomorphism from where s is the Sheffer sequence defined by  n n  t to the ring of shift-invariant operators  D 1  n n   t D snn Sp for each n  0 , and    t is the that maps Ss  n to SD  sn . In [8] n0 n! n0 n! unique formal power series such that  D . Finally is proved that given a delta operator D , and a series we also have the following characterization. n Proposition 2. Let sn  be a polynomial sequence. It   n1 n t  0  0 with 1  0 , then  D is n also a delta operator. Conversely, if  is a shift-in- is a Sheffer sequence if, and only if, there exists a delta operator D with basic set p such that variant operator (so that    D ), then if it is a n n delta operator, the unique series n n n sspnknkxy  x y  x,y. t  k  t k 0  n0 n n!  4. Umbral Composition such that  D   satisfies  0  0 and 1  0 . This section is based on [11]. 3. Sheffer Sequences Let p be a fixed polynomial sequence. Let us  n n n In this section, we also briefly recall some definitions define an operator  by  x  pn for each n  0 . Since p is a basis of  x , this means that  is and results from [8].  n n   Let pn be a sequence of polynomials in x . a linear isomorphism of  x . When p is the n0   n n We define the exponential generating function of basic set of a delta operator, then  is referred to as an pn as umbral operator, while if p is a Sheffer sequence, n0  n n tn then  is said to be a Sheffer operator. An umbral op- EGF pnn;t p  x t . n  n!  erator maps basic sets to basic sets, while a Sheffer op- n0 erator maps Sheffer sequences to Sheffer sequences. Let D be a delta operator and pn be its basic Let p be a polynomial sequence. For every n , n0  n n set. Let    t with   0 and   0 such n   0 1 pp xxkk where p xk is the coefficient that   D . Then from [8], nnk 0 of xk in the polynomial p   x . Let p and 1    n n EGF p ;t ext  . q be two polynomial sequences. Their umbral com- n n  n n position is defined as the polynomial sequence r   A polynomial sequence s is said to be a Sheffer n n n n0 pq# defined by  nnnn  sequence (also called a polynomial sequence of type zero n k in [9] or a poweroid in [10]) if there exists a delta opera- rpqnnk  x tor D such that k 0  for each n  0 . By simple computations, it may be 1) s0   , proved that rpqxxxkk n  . The set of 2) Dsnn1  n1 s for every n . nnk  all polynomial sequences becomes a (non-commutative) Following [9], a polynomial sequence sn is a n0 monoid under # with xn as identity. We observe Sheffer sequence if, and only if, there exists a pair  n

n that if T is the operator defined by Tqx  n for each Sheffer sequence is also unipotent. n  0 , then pq#  Tp. More generally, we It is clear from theorem 4 that the group of basic sets nnnn  n  n under umbral composition is isomorphic to the group of #k #k have Tqknx   where q is the substitutions. Moreover, the group of unipotent basic sets n n n n n also is isomorphic to the group of unipotent substitutions. k -th power of q for the umbral composition (it is n n Likewise, the group of (unipotent) Sheffer sequences is equal to a sequence say rk and we denote rk   n n n isomorphic to the group of (unipotent) substitutions with #k by qn ). Under umbral composition, the set of all pre-function (see also [12]). Sheffer sequences is a (non-commutative) group, called Lemma 1. Let ,  be a substitution with pre- the Sheffer group ([12]), and the set of all basic se- function, and let spnn ,   be the Sheffer sequence quences is a sub-group of the Sheffer group. nn and the basic set associated to the delta operator   From [8] we have the following result that combines   and the invertible shift-invariant operator   (this delta operators, basis sets, Sheffer sequences and umbral   means that p is the basic set of   , and composition.  n n  Theorem 1. Let Q and P be two delta operators 1 s   p for each n ). Then, , is a uni- with respective basic sets q and p . Let S nn  n n  n n and T be two invertible shift-invariant operators. Let potent substitution with pre-function if, and only if, s and t be the Sheffer sequences defined by psx1nn x for every n . n n n n nn s  Sq1 and tTp 1 for each n . Let , be nn nn Proof. Let us first assume that , is a unipotent two invertible series such that S  , T   . substitution with prefunction. We have p0 1 for every Let  , be two formal power series with  00 0   , basic set, so that p0 11 . Let n  . We have 110  such that Q    and P  . Then,   k . Then,  pnp1 is RT  P   is a shift-invariant opera-    k2 k nn1   equivalent to tor,  is a delta operator with basic sequence  nn1 pnpx1kk x. vqp # . Finally, let r be the Sheffer kk0=0nn1   nnnnnn    n n sequence given by rst # . Then, rRv 1 n  nnnnnn    nnBy identification of the coefficient of x on both sides, for each n   . we obtain np1x nn1  np 1x It may be proved that if s  is the Sheffer sequence nn1 n n (since  is assumed to be a unipotent substitution), and, obtained from the delta operator D   with basic n1 by induction, pn1 x1 . Besides, we have set  pn  and the invertible shift-invariant operator n 1 1 1 1 snn  p for each n . But   S  , i.e., snn Sp for each n , then the inverse (because there is a ring isomorphism between  t qn of pn with respect to the umbral composition 1 n n and   ), and  1t  , where    t . 1 is the basic set of the delta operator   , the inverse Then, by identification of the coefficient of xn , we have t of s with respect to the umbral composition nn n n n n spnnxx1  for every n . Conversely, let us 1 is the Sheffer sequence tqnn  . assume that sp, is the Sheffer sequence and    nnnn   5. Unipotent Sequences the basic set associated to the delta operator   and the invertible shift-invariant operator   with The basic set p of a delta operator D is said to be nn  n n psx1 x for every n . By construction we unipotent if the unique series  such that D    nn nn1 satisfies 1 1 (and, obviously,  0  0 ), i.e.,  is a have 11np1x nn  np 1x so that 1 1. unipotent substitution. A Sheffer sequence s asso-  n n nn Likewise, 0 spnnxx , so that 0 1. □ ciated to a delta operator D   (with  0  0 , 1  0 ) and an invertible shift-invariant operator 1 S   (with  invertible), i.e., snn Sp for 6. Generalized Powers of Unipotent every n where p is the basic set of D , is said to n n Substitutions with Pre-Function be unipotent if p is unipotent, and if  is unipo-  n n The purpose of this section is to define , for any tent, i.e., 0 1. It is also clear from the previous sec-   and any unipotent substitution with pre-function tion (theorem 4) that the (umbral) inverse of a unipotent ,  , and to prove that it is also a unipotent substitu- basic set is unipotent, and the (umbral) inverse of a tion with pre-function. Moreover we show that

 ,  is a one-parameter sub-group, i.e., n n skninixy  sk x p  k y   i0 i ,,,  for every ,  , and (2) n 0 n ki kni ,1,t .  TRxy,   i0 i Let , be a unipotent substitution with pre- Now, let z be a variable commuting with x and y , function of  t . Let p be the unipotent basic   n n and let us define set of the (unipotent) delta operator   . Let s   n n zzn  be the unipotent Sheffer sequence associated to   in in sLn zxxx,zii00  L and the (unipotent) invertible shift-invariant operator  . ii  n Let R be the umbral operator given by Rpx  n for and similarly, all n , and let T be the Sheffer operator defined by n zzn  Tsx  n for all n . It is easily checked that for every in in pNn zxxx,z  N, #k kn #k kn ii00ii integer k , pRn  x and sTn  x . In par-   nnn1 nkfor each n . As polynomials in the variable z , their ticular, for each n , Tsxxnn s xx k0 degrees are at most n . As polynomials in the variable (by Lemma 5). Therefore, TidL  , where z , sn zxy   , pn zxy   , nnkn1 Lsxxx  n for each n . The operator L is k 0 n n ppzx zy actually a lowering operator. Then according to section 2, i0 ini  i  it is possible to define TLL k    for  k0   and k  n n every   . Moreover, we have TT exp log   . spinzxi  zy i0 i  n  For each    , let us define sTn   x  for have also a degree at most n . Because the equations (1) every n . When  k  , we have sn k   and (2) hold for every integer k , the polynomials (in the kn # k variable z ) Tsx  . So that in this case, s k is the  n n n k n n unipotent Sheffer sequence associated to , . This pppnizxy  i0  zxni  zy i means that if  ,,    k , and q is the uni- kk  n n and potent basic set of the (unipotent) delta operator  k  , n n sspninizxy  zx  zy #k 1 i0 i then snkn q for each n . Similarly, let  nnn1 kk are identically zero, and the above equations hold for Rpxxnn k 0 p xx for every n . There- n1 every   . Therefore, p  is a polynomial nkk n n fore, RidN, where Npxxx  n is a k 0 sequence of binomial-type, and s  is a Sheffer n n lowering operator. Again for every   , we define sequence for every    . Moreover, for every  RRNN exp log k  . For each ,  , we have  k0   k pRRR xxnn    n n     , we define pRn   x for each n . In particular n kk #k Rpnn k=0 pxx R for  k  , pknn p , so that it is the basic set k  n ppxk of the unipotent delta operator   . Clearly,  k=0 nk  1 snknkpk   for each n . Thus for every so that ppq #  . Similarly, k  , we have  nnn nnn   sss   #  for every ,  . n n  nnnnnn    Moreover, pkninixy  pk x p  k y i0 i (1) pR0xxx00nn T n q 0. n  nnnn    n   n ki kni nn  RRxy,   i0 i Therefore,    p  and    s    n  n  n  n

Copyright © 2013 SciRes. AM L. POINSOT 17 are one-parameter sub-groups. It follows that REFERENCES #1 [1] A. Benhissi, “Rings of Formal Power Series,” Queen’s pp nnnn Papers in Pure and Applied Mathematics, Queen’s Uni- versity, Kingsone, 2003. and [2] R. P. Stanley, “Enumerative Combinatorics—Volume 1, #1 ss Volume 49 of Cambridge Studies in Advanced Mathe- nnnn matics,” Cambridge University Press, Cambridge, 2000. (inverses under umbral operation). [3] G. H. E. Duchamp, K. A. Penson, A. I. Solomon, A. Hor- We define ,  as the pair of formal power series zeal and P. Blasiak, “One-Parameter Groups and Combi-  , such that  is the substitution that defines natorial Physics,” Proceedings of the Symposium CO-    PROMAPH3: Contemporary Problems in Mathematical the delta operator   with basic sequence p  ,    n  n Physics, Cotonou, 2004, pp. 436-449. and  is the invertible series such that  [4] L. Poinsot and G. H. E. Duchamp, “A Formal Calculus on 1 the Riordan near Algebra,” Advances and Applications in spnn   Discrete Mathematics, Vol. 6, No. 1, 2010, pp. 11-44. for each    . Since p  and s  are n   n n   n [5] L. W. Shapiro, S. Getu, W.-J. Woan and L. C. Woodson, unipotent sequences, it is clear that  is unipotent, and “The Riordan Group,” Discrete Applied Mathematics, Vol. 34, No. 1-3, 1991, pp. 229-239.   is a unipotent substitution. It is also clear that doi:10.1016/0166-218X(91)90088-E whenever  k  , then  ,,    k . Let us kk [6] G. Markowsky, “Differential Operators and the Theory of check that     ,  is a one-parameter sub- Binomial Enumeration,” Journal of Mathematical Analy- group of the group of unipotent substitutions with pre- sis and Application, Vol. 63, No. 1, 1978, pp. 145-155. function. This means that for every ,  , doi:10.1016/0022-247X(78)90111-7 [7] G. H. E. Duchamp, L. Poinsot, A. I. Solomon, K. A. Pen-  ,,,        son, P. Blasiak and A. Horzela, “Ladder Operators and Endomorphisms in Combinatorial Physics,” Discrete Ma-   ,   thematics and Theoretical Computer Science, Vol. 12, No. 2, 2010, pp. 23-46. First of all, by definition,   is the unipotent substitution associated to the basic set [8] G.-C. Rota, D. Kahaner and A. Odlyzko, “Finite Operator Calculus,” Journal of Mathematical Analysis and Its Ap- ppp #  , plications, Vol. 42, No. 3, 1973, pp. 684-760. nnnnnn   [9] I. M. Sheffer, “Some Properties of Polynomial Sets of and therefore   . In a similar way, the se- Type Zero,” Duke Mathematical Journal, Vol. 5, No. 3, ries   is uniquely associated to the Sheffer sequence 1939, pp. 590-622. doi:10.1215/S0012-7094-39-00549-1 sss #  and to the basic set [10] J. F. Steffensen, “The Poweroid, an Extension of the Ma- nnnnnn   thematical Notion of Power,” Acta Mathematica, Vol. 73, ppp #  . Again this means nnnnnn   No. 1, 1941, pp. 333-366. doi:10.1007/BF02392231 that    . Therefore, we obtain the ex- [11] S. Roman, “The Umbral Calculus,” Dover Publications,   New York, 1984. pected result. It is also clear that 00,1,t   . [12] T.-X. He, L.C. Hsu and P.J.-S. Shiue, “The Sheffer Group Remark 2. In particular, since  is a field of and the Riordan Group,” Discrete Applied Mathematics, characteristic zero, for every q  , we may define Vol. 155, No. 15, 2007, pp. 1895-1909. q fractional powers , such as for instance n , doi:10.1016/j.dam.2007.04.006 for each integer n .