A STUDY OF UMBRAL CALCULUS
DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF Ma^ttv of $I)ilOjtfop()P IN APPLIED MATHEMATICS
BY NISAR K. S.
UNDER THE SUPERVISION OF PROF. MUMTAZ AHMAD KHAN M Sc , PhD (Lucknow)
DEPARTMENT OF APPLIED MATHEMATICS H. COLLEGE OF ENGINEERING 8c TECHNOLOGY ALIGARH MUSLIM UNIVERSITY ALIGARH-202002 U.P. (INDIA) 2008
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DS3601 PROF. MUMTAZ AHMAD KHAN M.Sc.Ph.D.(LUCKNOW) DEPARTMENT OF APPLIED MATHEMATICS ZAKIR HUSAIN COLLEGE OF ENGINEERING & TECHNOLOGY FACULTY OF ENGINEERING, ALIGARH MUSLIM UNIVERSITY ALIGARH - 202002, U.P. INDIA
cDated-:.Jj.\A'..?:^%. CERTIFICATE
Certified that the thesis entitled "^ Study of VmBraC CakuCus" is Based on a part of research wor^ 9ir. INisar 7(, S. carried out under my guidance and supervision in the (Department of Applied 'Mathematics, Vacuity of'Engineering, Adgarh MusCim Vniversity, JLdgarh. To the Best of my ^owCedge and BeCief the wor^incCuded in the thesis is originaC and has not Been suBmitted to any other Vniversity or Institution for the award of a degree.
This is to further certify that Mr. !Nisar % S. has fufiCCed the prescriBed conditions of duration and nature given in the statutes and ordinances ofJiCigarh !Musfim Vniversity, JLCigarh.
(PROF. MUMTAZ AHMAD KHAN) SUPERVISOR
Ph.No. 0571270027 Mob.No.09897686959 (Prayer
I Segin in the name ofjiCmighty ^od, the most gracious and the
Most 9/iercifuC without whose wiCC this thesis wouCd not have seen the dayCight.
M praise 6e to the JACmighty god, sustainers of the worfds, and may the Skssings and peace 6e upon the Master of'Prophets, on his progeny andatfhis companions.
The primary aim of education is to prepare and qualify the new generation for participation in the cutturat development of the people, so that the succeeding generations may Benefit from the
^owCedge and experiences of their predecessors.
CONTENTS
Page No.
Acknowledgement I - III Preface iv - V
Chapter-I INTRODUCTION 1-17 1. What is Umbral Calculus? 1 2. Historical Development of Umbral 3 Calculus 3. Applications of the Umbral Calculus 6 4. Generalization and Variants of the 12 Umbral Calculus
Chapter- II SHEFFER SEQUENCES 18-75 1. Introduction 18 2. Formal Power Series 23 3. Linear functionals 24 4. Linear Operators 28 5. Polynomial Sequence 32 6. Recurrence Formulas 36 7. Transfer Formulas 42 8. Umbral Composition and Transfer 44 Operators 9. Examples: Gegenbauer, Chebyshev 47 and Others 10 . Examples: Jacobi and Others 61 11 . Examples: The q-Case 66
Chapter - III EXTENSION OF UMBRAL 76 - 101 CALCULUS 1. Introduction 76 2. Decentralized Geometric Sequences 78 3. Decentralized Sheffer Sequences 80 4. Computation of Decentralized Sheffer 81 Sequences 5. The Classical Umbral Calculus 84 6. The Newtonian Umbral Calculus 89 7. Formal Power Series 94 Chapter-IV GENERALIZED UMBRAL 102--113 CALCULUS 1. Introduction 102 2. Formal Laurent Series 104 3. Linear Functionals 106 4. Linear Operators 109 5. Sheffer Sequence 110
Chapter-V A q-UMBRAL CALCULUS 114 -137 1. Introduction 114 2. An Algebraic Setting for Polynomials 119 of Binomial Type 3. The Umbral Lemma and Applications 125 4. Eulerian Families of Polynomial s 130 5. Homogeneous Eulerian Families 135
Bibliography 138 -163 AcknowCed^ement
All praises and thanks to the Almighty God who taught the man what he knew not and who blessed him wisdom and courage. It is his will that this dissertation has seen the daylight.
I would never have the words to express my gratitude and indebtedness to my supervisor Prof. Mumtaz Ahmad Khan, Ex-Chairman, Department of
Applied Mathematics, Aligarh Muslim University, Aligarh, under whose able guidance, valuable suggestions, encouragement, inspirations and motivations as well as his keen interest in my work that I am able to complete this work.
I shall always grateful to present Chairman Prof Mohammad Mukhtar
Ali for providing me necessary facilities in the department. I also take this opportunity to thank other faculty members of the department who have contributed in anyway towards the completion of this work.
I am thankful to my present Senior research workers namely Mr.
Bahman Alidad, Mr. Mohd. Asif and Mr. Mohd. Akhlaq pursuing their Ph.D. degree and to my fellow research workers Mr. Mohd. Khalid Rafat Khan, Mr.
Isthrar Ahmad and Mr. Shakeel Ahmed pursuing their M.Phil, degree under the supervision of my supervisor.
I also take this opportunity to remember my past Senior research workers namely Dr. Abdul Hakim Khan, Dr. Mrs. Muqaddas Najmi, Dr. Ajay
Kumar Shukla, Dr. Khursheed Ahmad, Dr. Ghazi Salama Mohammad Abukhammash, Dr. Abdur Rahman Khan, Dr. Bhagwat Swaroop Sharma, Dr.
Mukesh Pal Singh, Dr. Bijan Rouhi, Dr. Naseem Ahmad Khan and Dr. R. P.
Khan all of whom obtained their Ph.D. degree and Mr. Syed Mohd. Amin, Mr.
Ajeet Kumar Sharma, Mr. Shakeel Ahmad Alvi, Mr. Khursheed Ahmad, Mr.
Bhagwat Swaroop Sharma, Mr. Mohd. Asif and Mr. Mohd. Akhlaq who obtained their M.Phil, degrees under the supervision of my supervisor Prof
Mumtaz Ahmad Khan.
My thanks are also due to other research fellows of the department Mr.
Waseem Ahmad Khan, Mr. Syed Mohd. Abbas, Mr. Naeem Ahmad, Mr.
Manoj Singh, Miss. Naghma Irfan, Miss Tarannum Kashmin and Miss Rabiya
Katoon.
I also express my thanks to all non-teaching staff members of the
Department of Applied Mathematics, who have contributed in one way other towards my work.
Most of the credit of my work would be to my loving parents Mr. K. K.
Sooppy and Mrs. Aleema whose sustained patience, prayers and blessings on me enabled me to complete this task.
I shall be failing in my duties if I do not express my gratitude to the wife of my supervisor Mrs. Aisha Bano (M.Sc, B.Ed.) T.G.T., Zakir Hussain Model
Senior Secondary School, Aligarh for her kind hospitality and prayer to God for my welfare. I am thankful to my supervisors children Miss. Ghazala
Parveen Khan (Daughter) M.Tech. (Electrical), Mr. Rashid Imran Ahmad Khan (Son) B.Tech. (Petro-Chemical) and to 15 year old master Abdul Razzaq Khan
(Son) a class of X student of Our Lady of Fatima School, Aligarh for enduring the preoccupation of their father with this work.
I also owe a few words of my friends Mr. Saiful Rahman Mondal (I.I.T.,
Roorkie), Mr. Akhlaq (I.I.T., Kanpur), Mr. Javed AH (Department of
Mathematics, A.M.U.) and Mr. Mahamudal Hasan for their good wishes and encouragement.
At the end I would like to thank Mr. I. A. Farooqi, Dr. V. P. Singh, Adv.
Naubat Singh and Mrs. Sunita Singh for their cooperation and help.
Finally I owe a deep sense of gratitude to the authorities of Aligarh
Muslim University, my Alma mater, for providing me adequate research facilities.
(NISAR K. S.)
Ill (Pr^ace
Umbral calculus provides a formalism for the systematic derivation and classification of almost all classical combinatorial identities for polynomial sequences, along with associated generating functions, expansions, duplication formulas, recurrence relations, inversions, Rodrigues representation, etc.
The dissertation comprises of five chapters. The first chapter covers a comprehensive account of the historical origin of Umbral calculus and the upto date developments made in these areas. This chapter also contains application
of Umbral calculus, Generalizations and variants of Umbral calculus.
In II chapter formal Power series. Linear functionals, Linear operators,
Polynomial sequence. Recurrence and Transfer formulas are given. This
chapter also contains Umbral composition Transfer operators and various
examples of Gegenbauer, Chebyshev, Jacobi and others. Besides, some
examples of q-analogues of certain polynomials are also included in it.
Chapter III deals with the extension of Umbral calculus. The chapter
contains Decentralized Geometric Sequences, Decentralized Sheffer
Sequences, Computation of Decentralized Sheffer Sequence, The Classical
Umbral Calculus, The Newtonian Umbral Calculus and the Formal Power
Series.
IV In chapter IV, we present some Generalization of Umbral Calculus. This chapter contains Formal Laurent Series, Linear Functionals, Linear Operators and Sheffer Sequences.
The V and final chapter is about "A q-Umbral Calculus".. Besides, the chapter contains an algebraic setting for polynomials of Binomials type. It also contains the Umbra lemma and applications, Eulerian families of polynomials,
Homogeneous Eulerian families etc.
The dissertation is in concluded with a bibliography which by no means is exhaustive one but lists only those books and papers which have been referred to in this dissertation. CHAPTER -1
INTRODUCTION CHAPTER-I
INTRODUCTION
WHAT IS UMBRAL CALCULUS?
The theory of binomial enumeration is variously called the calculus of finite differences or the Umbral calculus. This theory studies the analogies between various sequences of polynomials Pn and the powers sequence x". The
subscript n in pn was thought of as the shadow ("Umbra" means "Shadow" in
Latin, whence the name umbral calculus) of the subscript n in x", and many
parallels were discovered between such sequences.
Take the examples of the lower factorial polynomials (x)n = x (x - 1)
(x - n + 1). Just as x" counts the number of functions form an n-element
set to x-element set, (x)n counts the number of injections, just as the derivative
maps x" to nx"~', the forward difference operator maps (x)n to n(x)n-i. Just as
also polynomials can be expressed in terms of x" via Taylor's theorem
.( . ^ a"D"f(x) f(x + a)=X -^, n=o n!
Newton's theorem allows similar expression for (x)n
n=o n! where Af(x)=f(x + l)-f(x). Just as (x + y)" is expanded using the binomial theorem (x + a)"=^ a'' x""'' ,(x + y)n expands by vandermonde's k=0 vk/ identity
^n>
k=0 V^/
For further such identities see [146, 152]
This theory is quite classical with its roots in the works of Barow and
Newton-expressed in the belief the some polynomial sequences such as (x)n really were just like the powers of x. Nevertheless many doubts arose as to correctness of such informal reasoning, despite various [48] attempts to set it on an axiomatic base.
The contribution of Rota's school was to first set umbral calculus on a firm logical foundation by using operator methods [152, 190]. That being done, sequences of polynomials of binomial type
II ^n^ Pn(x + y)=Z Pk(x)Pn-.(y) (1) k=0 vk; could be for once studied symmetrically rather than as a collection of isolated yet philosophically similar sequences. The sister sequence of divided powers
q^ (x) = i-^^-^ then obeys the identity n!
n qn(>^ + y)=Z qk(x)qn-k(y)- (2) k=0 Given any species of combinatorial structures (or quasi-species), let
Pn(x) be the number of functions from an n-element set to x-element set enriched by this species. A function is enriched by associating a (weighted) structure with each of it fibers. All sequences of binomial type arise in this manner and conversely all such sequences are of binomial type.
HISTORICAL DEVELOPMENT OF UMBRAL CALCULUS
The history of the Umbral calculus goes back to the IV"' century. The rise of the Umbral calculus, however, takes place in the second half of the 19'^ century with the work of such mathematicians as Sylvester (who invented the name) Cay ley and Blissard [46]. Although widely used, the Umbral calculus was nothing more than a set of "magic" rules of lowering and raising, indices
[128]. These rules worked well in practice, but lacked a proper foundation. Let us consider an example of such a "magic rule". The Bernoulli numbers B„ are defined by the generating function
IB„^=^ (3) n=o n! e"-l The magic trick used in the 19"^ century Umbral calculus is to write
CO n 00 n iB.^^XB-^-e- (4)
n=o n! n=o n! where we use the i ^ symbol to stress the purely former character of this manipulation. A trivial standard algebraic manipulation then yields e^'^^')'' - e-^^ « X (5)
x" from which we deduce by equating coefficients of — that n!
(B + l)" - B"« 5,, (6) where b^^ denotes the Kroneker delta. If we now expand (6) using the binomial theorem and change the superscript back to subscripts, we obtain the following relation for the Bemaulli numbers:
n-l n I B. =5,„ (7) k=0 vky which can be shown to be true (a standard direct proof is possible by
^e^-1^ considering the reciprocal power series V "^ J
Early attempts to put the Umbral manipulation on an axiomatic basis
[48] were unsuccessful. Although the mathematical world remained sceptical of the umbral calculus, it was used extensively.
A second line in the history of umbral calculus in the form that we know today, is the theory of Sheffer polynomials. The history of Sheffer polynomials goes back to 1880 when Appell studied sequences of polynomials (pn)n satisfying p'^ = n p„_, [22]. These sequences are nowadays called Appell polynomials. Although this class was widely studied [114], it was not until
1939 that Sheffer noticed the similarities with which the introduction of this survey starts. These similarities led him to extend the class of Appell polynomials which he called polynomials of type zero [198], but which nowadays are called Sheffer polynomials. This class already appeared in [147].
Although Sheffer uses operators to study Sheffer polynomials, Sheffer polynomials is mainly based on formal power series. In 1941 the Danish actuary Steffensen also published a theory of Sheffer polynomials based on formal power series [206]. Steffensen uses the name poweroids for Sheffer polynomials [200, 201, 206, 207, 208, 209]. However, these theories were not adequate as they do not provide sufficient computational tools (expansion formulas etc.)
A third line in the history of the Umbral calculus is the theory of abstract linear operators. This line goes back to the work of Pincherle starting in the
I980's. Pincherle's early work is laid down in the monumental monograph.
Pincherle went surprisingly for considering the state of functional analysis in those days, but his work lack explicit examples. The same applied to papers by others in this field [112, 114, 222].
A prelude to the merging of these three lines can be seen in [190], in
which operators methods are used to free umbral calculus from its mystery. In
[152] the ideas from [190] are extended to give a beautiful theory combining
enriched functions, umbral methods and operator methods. However, only the
subclass of polynomials of binomial type are treated in [152]. The extension to
Sheffer polynomials is accomplished in [193]. The latter paper is much more
geared towards special functions, while the former paper is a combinatorial
paper. The papers [152] and [193] were soon followed by papers that reacted directly on the new Umbral calculus. E.g., Fillmore and Williamson showed that with equal case the Rota umbral calculus, could be situated in abstract vector spaces instead of the vector space of polynomials [123], Zeilberger noticed connections with Fourier analysis [233] and Garsia translated the operator methods of Rota back into formal power series [124].
We conclude this section with mentioning the remarkable papers [98,
122, 194, 195, 196] in which the authors manage to make sense of the 19" century Umbral calculus theory fulfilling Bell's dream [46].
APPLICATIONS OF THE UIMBRAL CALCULUS
We now indicate papers that apply the umbral calculus to various fields.
Lagrange Inversion
An important property the umbral calculus is that it has its own generalization of Lagrange's inversion formula (as follows from the closed forms of basic polynomials [152], in particular transfer formula). Thus we find many papers in which none forms of Lagrange's inversion formula is derived using umbral calculus [26, 39, 109, 130, 205, 215, 221]. Symmetric Functions
In [144], the umbral calculus is generalized to symmetric functions when counting enriched functions (functions, injections, reluctant functions, dispositions, etc.) from N to X, we can assign a weight to each function according to its fiber structure.
w(f) = HigN f (0 = HxeN ^ " T^he total number of such functions is a symmetric function Pn(x) of degree n where n = INI. The elementary and complete symmetric functions are (up to a multiple of n !) good examples of such sequences. They obey their own sort of binomial theorem
11 ^n^ Pn(x^y)=Z Pk(x)Pn-k(y)- k=0 vky
The generating functions of pn(x) are directly related to that of their underlying species. By specializing all x variables to x, we return to the study to polynomials.
Nevertheless, such a sequence pn(x) is not a basis for the vector space of symmetric functions. Furthermore, the Pn(x) may not even be algebraically independent. These problems were solved in [148, 149] where an Umbral calculus of full sequences of symmetric functions pn(x) indexed by an integer partition A, is presented. Combinatorial Counting and Recurrences
Another rich field of application is linear recurrences and lattice path counting. Here we should first of all mention the work of Niederhausen [156,
157, 158, 159, 160]. The starting idea of the work of Niederhausen is the fact that of Q is a delta operator, the E^Q is also a delta operator, and hence as basic sequence. The relation between the basic sequences of these operators enables to him to upgrade the binomial identity for basic sequences to a general Abel
like identity for Sheffer sequences. For a nice introduction to this, 1 refer to the
survey papers [158, 159]. Inspite of their titles, the papers [225, 226] are more
directed to the general theory of umbral calculus, then to specific applications
in lattice path counting. A different approach to lattice path counting is taken in
the papers [171, 188]. In these papers a functional approach is taken in the spire
of [189, 182] rather than an operator approach. Finally, an approach based on
umbra can be found in [221]. Umbral calculus is strongly related with the Joyal
theory of species [105, 197].
Graph Theory
The chromatic polynomial of graph can be studied in fruitful way using
a variant of Umbral calculus. This is done by Ray and Co-workers [142, 164,
169, 170]. A generalization of the chromatic polynomial to so called partition
sets can be found in [141]. Coalgebras
Coalgebraic aspects of Umbral calculus are treated in [100, 101, 142,
162, 168]. E.g. Umbral operators are exactly coalgebra automorphisms of the usual Hopf algebra of polynomials.
Statistics
Non parametric statistics (or distribution free statistics) has a highly combinatorial flavor. In particular, lattice path counting techniques are often used. It is therefore not surprising that the main applications of Umbral calculus to statistics are of combinatorial nature [153, 154, 155, 157].
However, there are also applications to parametric statistics. Di
Bucchianico and Loeb link natural exponential families with Sheffer polynomials. They show that the variance function of natural exponential families determines the delta operator of the associated Sheffer sequence. As a side result, they find all orthogonal Sheffer polynomials. Another application concerns statistics for parameters in power series distributions [103, 138].
Probability Theory
As suggested in [193], there is a connection between polynomials of binomial type and compound Poisson process. Two different approaches can be
found in [97. 204]. A connection of polynomial of binomial type with renewal sequence can be found in [203]. Probabilistic aspects of Lagrange inversion and polynomials of binomial type can be found in [205]. Various probabilistic representations of Sheffer polynomials can be found in [116]. Many of the above results can also be found in the book [117].
Topology
Application of umbral calculus to algebraic topology can be found in the work of Ray [162, 163, 165, 166, 167]. An application of umbral calculus to
(co) homology can be found in [139, 140].
Analysis
Cholewinski developed a version of the Umbral calculus for studying differential equations of Bessel type and related topics in [107] and [107].
A connection between approximation operators and polynomials of binomial type can be found in [132]. Further papers in this direction are [133,
150,210].
Orthogonal polynomials play an important role in analysis. It is therefore important to know whether polynomials are orthogonal. The classification of orthogonal Sheffer polynomials was first found by Meixner
[147]. Orthogonal Sheffer polynomials on the unit circle have been characterized by Kholodov [134]. General papers on orthogonal polynomials
and umbral calculus are [125, 134]. Hypergeometric and related functions are
delta with in an Umbral calculus way in [212, 213, 232]. There are different ways of implementing q-analysis in terms of umbral calculus. The first q-umbral calculus can be found in [18]. Other q-umbral calculi can be found in [5, 8, 108, 109, 110, 131, 136, 172, 183]. Comparison between different q-Umbral calculi can be found in [4, 183]. Various basic hypergeometric i.e. q-hypergeometric identities are derived in [104]. A q-
Saalschatz identity is derived in an umbral way [202].
f( \—f( ) Constructing umbral calculi based on the operator f(x)^-^^-^—^ x-y yields a powerful way to study interpolation theory [177, 200 and 216].
Banach algebras are used by Di Bucchianico [115] to study the convergence properties of the generating function of polynomials of binomial type and by Grabiner [126, 127] to extend the umbral calculus to certain classes of entire functions.
Application of Umbral calculus to numerical analysis can be found in several papers of Wimp [228, 229, 230, 231].
Umbral calculus is a powerful tool for dealing with recurrences.
Recurrences play an important role in the theory of filter banks in signal processing. An umbral calculus approach based on recursive matrices can be found in [35, 36]. A related theory is the theory of wavelets. An umbral approach to the refinement equations for wavelets can be found in [199].
11 Physics
An application of umbral calculus to the physics can be found in [227].
Biedenharm and his Co-workers use umbral techniques in group theory and quantum mechanics [53 and 54].
Gzyl found connection between umbral calculus, the Hamiltonian approach in physics and quantum mechanics [129].
Morikawa developed an umbral calculus for differential polynomials in infinitely many variables with applications to statistical physics [151].
Invariant Theory
There are some papers that link invariant theory (either classical or modern forms like super symmetric algebras) with umbral calculus [73, 102].
GENERALIZATION AND VARIANTS OF THE UMBRAL
CALCULUS
The umbral calculus of [193] is restricted to the class of Sheffer polynomials. It was therefore natural to extend the umbral calculus to larger classes of polynomials. Viskov first extended the umbral calculus so-called generalized Appell polynomials (or Boas-Buck polynomials) [219] and then went to generalize this to arbitrary polynomials [220]. The extension to generalized Appell polynomials makes it possible to apply umbral calculus to q-analysis or important classes of orthogonal polynomials like the Jacobi
12 polynomials [183]. Roman remarks [182] that Ward way back in 1936 attempted to construct an umbral calculus for generalized Appell polynomials
[222].
An extension of the Umbral calculus to certain classes of entire function can be found in [126, 127].
Another extension of umbral calculus is to allow several variables [25,
80, 133, 174, 223, 224]. However, all these extensions suffer from the same drawback, viz. they are basis dependent. A first version of basic free umbral calculus in finite and infinite dimensions was obtained by Di Bucchianico,
LoebandRota[12].
Roman [178. 179, 181, 182] developed a version to the umbral calculus
for inverse formal power series of negative degree most theorems of umbral
calculus have their analog in this context. In particular, any shift invariant
operator of degree one (delta operator) has a special sequence associated with it
satisfying a type of binomial theorem. Nevertheless, despite its philosophical
connections, this theory remained completely distinct from Rota's theory
treating polynomials.
Later, in [146], a theory we discovered which generalized
simultaneously Roman and Rota's umbral calculus by embedding them in a
logarithmic algebra containing both positive and negative powers of x, and
logarithms. A subsequent generalization [143, 145] extends this algebra to a
field which includes not only x and log (x) but also the interacted logarithms,
all of whom may raised to any real power. Sequences of polynomials Pn(x) are
13 then replaced with sequence of asymptotic series p^ where the degree a is real and the level a is a sequence of real. Rota's theory is the restriction to level a =
(0) and degree a e N. Roman's theory is the restriction to level a = (1) and degree a 6 Z. Thus the difficulty in uniting Roman's and Rota's theories was essentially that they lay on different levels of some larger yet unknown algebra.
Rota's operator approach to the calculus of finite difference can be thought of as a systematic study of shift-invarient operators on the algebra of polynomials. The expansion theorem [152] states that all shift-invariant operators can be written as formal power series in the derivative D.
If 0: C[x] -> C[x] is a shift invariant operator, then
k=o n! where a„=[ex"^
However a generalization of this Kurbanov and Maksimov [137] to arbitrary linear operators has received surprisingly little attention. Any linear operator 6: C[x]-> C[x] can be expand as a formal power series in x and D is the operator of multiplication by x. More generally, let B be any linear operator which reduces the degree of non-zero polynomials by one (By convention, deg
(0) = - 1). Thus, B might be not only the derivative or any delta operator, but also the q-derivative, the divided difference operator, etc. Then B can be expanded in terms of x and B: e=Sf„(x)B". k=0
A detailed study of this kind of expansion and its sister expansion
00 6 = ^ f„(B)x" can be found in [120]. Extensions of umbral calculus to k=0 symmetric functions have already mentioned. Another interesting extension is the divided difference Umbral calculus, which is useful for interpolation theory. Extension of the umbral calculus to the case where the base field is not characteristic zero [214, 217, 218].
An important variant is the umbral calculus that appears by restricting the polynomials to integers. This theory is developed by Bamabei and
Co-authors [28, 29, 30] and is called the theory of recursive matrices. There are applications in signal processing [35, 36] and to inversion of combinatorial sums [111].
The theory of umbral calculus dates back to series of three papers of
Steven Roman [178, 179, 181]. The underline theme of the paper due to
S. Roman [178] is that special polynomials sequences which he has form
Sheffer sequences possess a unified theory. Form this theory many seemingly unrelated results in the classical literature demarche as special cases of general results. Moreover, the umbral calculus provide a cohesive approach to the study of further properties of the sequences, as well as bringing to light some new sequences which are strongly related to the important classical once. In fact in his paper in 1982, S. Roman [178] studied a certain class of polynomial sequences, called Sheffer sequences, by studying the dual vector space p to the algebra polynomials p. First we made p into an algebra - the algebra of formal power series in t. Sheffer sequences were defined as those sequences in p which are orthogonal to some geometric sequence g(t) f(t) in
g(t)f(t)S„(x)) = c„5„,,
The degree requirements deg g(t) = 0 and deg f(t) = 1 were placed in order to ensure the existence and uniqueness of the sequence Sn(x).
In his paper, 1982, S. Roman [179] extended the theory to a more general class of sequences in p than the class of geometric sequences. This brought some important new polynomial sequences into the purview of the umbra! calculus. He called elements of this new class decentrahzed geometric sequences.
In his paper, 1983, S. Roman [181] replaced the algebra o of polynomials in x by the field A of formal Laurent series in x of the form
m P(x) = l b^ x^ for any integer m. The algebra of formal power series in t. Which gave him his representation of linear functional and linear operators on p is replaced by the field r of formal Laurent series in t of the form
16 f(t)-Ia, t'' k=n
Then F is used to represent all the continuous linear functional and some continuous linear operators on A in much the same manner as in [178].
The aim is to further develop the theory of umbral calculus to bring in its purview the other polynomials also. CHAPTER - II
SHEFFER SEQUENCES CHAPTER - II
SHEFFER SEQUENCES
1. INTRODUCTION
The present chapter deals with the development of the modem theory of the Umbral calculus.
At the very outset a brief explanation of the term modern umbral calculus is given. A large part of applied analysis is concerned with the study of certain sequences of special polynomials. Some of the most important of these sequences are associated with the names of Jacobi, Gegenbauer,
Legendre, Chebyshev, Bessel, Laguerre, Hermite and Bernoulli. All of these sequences and many more, fall into a special class. Boas and Buck, in their work on polynomial expansions of analytic functions, used the term sequence of generalized Appell type for members of this class. A sequence pn(x) of polynomials is of generalized Appell type if it has a generating function of the form
A(t)4>(xh(t))^XP.Wt^ 1(=0 where
00
k=0
"^(0=1; "^kt"" 'i'^.^O for all k, k=0 h(t)=Xh,t'' h,;.0, k=0
The modem umbral calculus grew out of an attempt to develop a unified theory for this class of polynomial sequences. S. Roman and G.C. Rota [189] developed the theory to deal effectively with an important subclass known as the sequences of Sheffer A-type zero, whose generating functions are of the form
A(t)e'^<')=j; hMi\ (1.1)
k=0
The subclass includes the important sequences of Hermite, Laguerre and
Bernoulli. We remark that the sequence pn(x) is normalized by the presence of k! on the right side of (1.1). Extension of the theory to the entire class of generalized Appell sequences is dealt here.
Section 2 of the present chapter contains a review of needed facts about formal power series. Section 3 discusses the dual vector space P* of all linear functionals on the algebra P of polynomials. For it is in the umbral calculus that one studies the algebra P via its dual space P*. In this section the structure of an algebra is put on the vector space p*. Then since we may multiply linear functionals, the notion of a geometric sequence ML for k = 0, 1, 2, and
M and L is P* makes sense. Section 4 defines a certain algebra of linear operators on P which is isomorphic to the algebra P*. The notational convenience of the linear operator proves indispensable to the theor>'. In
Section the main object of study - the Sheffer sequence is given, a sequence
19 Sn(x) of polynomials is the Sheffer sequence for a pair of linear functionals (M,
L) if it is orthogonal to the geometric sequence ML*', that is, if
(ML^|S„(X)) = CA,. for all n, k > 0 where Cn is a fixed sequence of non-zero constants and the notation (N|P(X)) is used for the action of N in P* on p(x) in P, where 5n,k is the Kronecker delta function 5n,k = 0 if n ^ k and 5n,n = 1. In this chapter we give several characterizations of Sheffer sequences, including the generating function
A(t)e.(h(t)) = i ^t\ (1.2) k=0 where A(t) and h(t) are as before and
is a generalization of the exponential series (e^ (t) = e"' if Cp = n !). Thus we see
that if ^n = 1/Cn, then the sequence Sn(x)/Cn is of generalized Appell type as
defined by Boas and Buck [62]. Also included in Section 5 is an algebraic
characterization of Sheffer sequences which may be thought of as
generalization of the binomial formula. In Section 6 we derive recurrence
formulas for Sheffer sequences and in Section 7 we give a powerful formula
for the direct computation of Sheffer sequences. Section 8 is devoted in to the
connection - constants problem of determining the constants &„\^ in
CO s„(x)=X 3n,kPk(x), k=0
20 where Sp (x) and Pn (x) are given Sheffer sequences.
Next, consider one of the most innovative aspects of the present theory.
Suppose Sn(x) is the Sheffer sequence for the pair of Hnear functionals (M, L) where M is not the multiplicative identity in the algebra P*. If we denote this identity by e, then the Sheffer sequence pn(x) for the pair (s, L) bears a strong association to Sn(x). Many of the properties of Sn(x) are possessed by pn(x) and yet in some sense pn(x) is a simpler sequence. Now the point is that almost all of the well-known classical sequences are of the type Sn(x). In the Hermite and
Bernoulli cases the associated sequence pn(x) is the simple sequence x". In the
Laguerre cases L*"'(x) of order a, the simpler associated sequence is the
Laguerre sequence L*^''(x). But up to now there had been no clue to the existence of such sequences pn(x) associated to, for example, the Jacobi,
Gegenbauer or Chebyshev sequences. A major portion of the examples is devoted to the study of the properties of these new sequences.
For the Gegenbauer polynomials Gn(x) Rainville sets
(i-2xt+t^r=j;G„(xK k=0
In Section 9 the Sheffer sequence Sn(x) characterized by
(i-ixt-ft^r-x (x)t' k=0 vky have been studied.
21 r-^' Thus s„ (x) is the Gegenbauer sequence. For the Chebyshev k. polynomials of the first kind we have
(l-x^)(l-2xt + t^r=l + 2X T,(x)tV
In Section 9 the Sheffer sequence characterized by (i-xO(i-2xt-ftO"' = i: (-irs.(x)t^ k=0 have been studied.
A similar normalization factor is required for the Chebyshev polynomials of the second kind. Finally, for the Jacobi polynomials Pn(x) Rainville [161] gives the generating function
l + g + P 2 + a + p -l-a-p ; 2t(x-l) (i-t) 2F1 (1-t)^ 1 + a;
k=o {\ + af
In Section 10 the Sheffer sequence Jn(x) satisfying
l + g + p 2 + a + (3 l-a-p 2xt (i-t) 2F, 2 2 1 + a; (1^
(l + a + + a + p) x(k) ^ 2 J,(x)t^ z M k=0 (Ua)' have been studied.
22 From this one can easily obtain Jn(x) in terms of the classical Jacobi polynomials.
2. FORMAL POWER SERIES
In this section we give a few basic facts about formal power series. Let
F be the algebra of all formal power series in the variable t over the field K
(characteristic zero). Addition and multiplication in F is purely formal and F is well known to be an integral domain. If
f(t) = Za,t\ (2.1) k=0 then the degree of f(t) is the smallest k such that a^ ^ 0. It is easy to see that deg f(t) g(t) = deg f(t) + deg g(t).
These series f(t) has a multiplicative inverse in F, denoted by f '(t) and l/f(t), if and only if deg f(t) = 0. Such a series is called invertible.
Suppose gk(t) is a sequence in F for which deg gi<(t) > k. Then if a^ is a sequence of constants, the sum
CO
k=0 is a well-defined series in F, found by simply collecting coefficients of like powers of t. In particular, we may take gk(t) = g(t)'^ where deg g(t) > 1.
If f(t) is given by (2.1), we may form the composition
f(g(.))=Ia.g(.)' k=0
23 which is a well-defined element of F provided deg g(t) > 1. It is clear then deg f(g(t)) = deg f(t). deg g(t).
The series f(t) has a compositional inverse, denoted by f(t) and satisfying f(f(t)) = f(f(t))= t, if and only if deg f(t) = 1. We call any series f(t) with deg f(t) = 1 a delta series.
A sequence gk(t) for which deg gk(t) = k forms a pseudo basis for F. In other words, for each series f(t) there is a unique sequence of constants for which
f(t)=Za,g,(t). k=0
In particular, the powers of a delta series form a pseudo basis for F.
3. LINEAR FUNCTIONALS
Let P be the algebra of polynomials in a single variable over K and let
P* be the dual vector space of all linear functionals on P. We use the notation
(L|P(X)) for the action of L in P* on p(x) in P. Any linear functional L in P* is uniquely defined by specifying the value /L X" \ for n > 0.
Let Cn be a fixed sequence of non-zero constants. We use this sequence to define, for each f(t) in F, a linear functional in P* as follows. If
f (t) = X!r=o ^k^*'' then the linear functional f(t) satisfies
f(t)k") = c„a„ (3.1)
24 for all n > 0. Notice that we have used the same notation f(t) for the power series and the linear functional. This should cause no confusion since if f(t) and g(t) are in F, then f^t) = g(t) if and only if (f (t)|x") = (g(t)|x") for all n > 0. In other words, f(t) and g(t) are equal as formal series if and only if they are equal as linear functionals.
The action defined in (3.1) depends on the particular choice of the sequence Cn.
As a consequence of (3.1) we have
x"> = c„8„. and
Za,t^ X";; \k-0 k=0 and so for any p(x) in P,
Za,t^ P(x) =Za,(.'|p(x); \k=0 / k=0
Now any linear functional L in P* can be represented as a series in F. In fact, if
.(0=1 ^^ (3.2) k=0 CK then
f,,(t)|x") = I k=0 C|,
25 -m and so as linear functionals ftCt) = L.
It is easily verified that the map L -> fi (t) is a vector space isomorphism from P* onto F. We shall obscure this map by identifying P* as the vector space F of all formal power series in t. Thus from now on we shall write our linear functionals in the form of power series in t.
The isomorphism L -> i(ff) has induced a natural product on linear functionals-namely, that of formal power series. In symbols fLM(t) = fiit) fM(0-
Some simple consequences of these results.
Proposition 3.1: If def f(t) > deg p(x), then
{f(t)|p(x)) = 0
Proposition 3.2: If f(t) is in F, then
f(t)=I ^^-^L/t^ (3.3)
k=o c,,
Corollary 1: If deg Pn(x) = n and {f(t)|p„(x)) = 0 for all n > 0 then f(t) = 0.
Proposition 3.3: If p(x) is in P, then
p(x)=Z -^-^ -^ • (3-4) k>0 C|(
Corollary 2: If deg fk(t) = k and (f„ (t)|p(x)> = 0 for all k > 0, then p(x) = 0.
Proposition 3.4: If f(t) is a delta series and
g(t)-Za,f(ty , k=0
26 then
(g(t)|x") = Ia,(f(tf|x"). k=0
One of the most important linear functionals on P is the evaluation functional denoted, for y in K, by Ey(t) and defined by
{s,(t)|p(x)> = p(y).
In view of (3.3) we have
00 .,k
k=0 ^k
It is interesting to note the form of Sy(t) for various choices of the sequence c^
For example, if Cp = n !, then
1 and of c. a^ then V'V
ey(t) = (l+ytf.
It will be convenient to introduce a linear operator 5t on F by setting
at'=^t'^', k> 'k-i k = 0 and extending by infinite linearity. Then we have for all n > 0 and k > 0,
27 x-x") = (t^|x"^')
~^n+l "n+l,k
'k-l
^k ^k-1
'k-l
= 3.t" X").
Proposition 3.5. If f (t) is in F, then
(5,f(t)|p(x)) = (f(t)|xp(x)) for all p (x) in P.
As a final remark, when we are thinking if a delta (or invertible) series
f(t) as a linear functional we shall refer to it as a delta (of invertible) functional.
4. LINEAR OPERATORS
If f (t) in F has the form f (t) = ^" a^ t'', then we shall define the
linear operator f (t) on F by
f(t)x"=X -^a, X"- (4.1) 1,-n n-k
Since f (t) and g (t) are equal as formal power series if and only if they are
equal as linear operators. [To see the "if part take successively n = 0, 1,2, ...
in (4.1)].
Notice that we are using juxtaposition to denote the action of an operator
on a polynomial. A little practice will remove the discomfort involved in
28 thinking of an element f (t) in F as either a formal power series, a linear functional or a linear operator, and the notational difference between
(f(t)ip(x)) and
f(t)p(x) will make the particular type of action of f (t) on p (x) clear.
The action f (t) p (x) depends on the sequence c^. However, we shall think of this sequence as being fixedan d so no confusion should arise.
It follows from (4.1) that
t'' x" -^ x""', n>k
= 0, n > k and so
^n ^"-J x"~''"^ C C , n-j n-k-J
= -^ t" x"-J
= t'(t^x").
Therefore, if f (t), g (t) are in F we have
[f(t)g(t)]p(x)=:f(t)[g(t)p(x)] (4.2) and we may write f (t) g (t) p (x) without ambiguity.
Notice also that
29 F(t)g(t)p(x) = g(t)f(t)p(x).
Actually, (4.2) shows that the product in Fin composition of operators.
When we are thinking of a delta (or invertible) series f (t) as an operator we may refer to it as a delta (or invertible) operator.
The relationship between the linear functional f (t) and the linear operator f (t) is given in the next theorem.
Theorem 4.1: If f (t), g (t) are in F, then
(g(t)f(t)lp(x)) = (g(t)|f(t)p(x)) for all p (x) in P.
For many choice of the sequence Cn, the operator t may be expressed in terms of some more familiar operators. To fix the notation we use
(x)n-x(x-l)...(x-n+l),x^"^ = x(x+l)...(x + n-l)
Dx" =nx"-',
D"' v" - x""^' U \ — A i n + 1 x-'x" =x"-', x-'l = 0 .
Then
(1) whencn = n!, t = D;
(2) whencn= 1, t = x ;
(3) whencn = (n!)'"^', t = (Dx)'^D;
(4) whenc„=-p-^, t = -(?. +XD)"'X-';
(5) whenc„=-, t = x"'D"'x"'; n
30 (6) when c„ = y\^, t = -{X + XD)"' D;
V" J
t = 4(1 + a + p + 2XD)"'(2 + a + p + 2XD)~'X"'(a + XD);
(S) when c. J'-f •('-<).
tp(,)Jp(qx)-p(x)) (qx - x)
Form the point of view of the present theory the operator t is the natural operator for studying various polynomial sequences. Case 1 is related to sequences of Sheffer A-type zero, such as the Hermite and Laguerre polynomials. This case has been studied by S. Roman and G. C. Rota [189].
Case 6 concerns itself with such polynomials as those of Gegenbauer and
Chebyshev. Case 7 relates to the Jacobi polynomials and Case 8 is the so called q - case.
Now we discuss some remarks concerning the series Sy (t). This series acts as the same linear functional, namely, evaluation at y, regardless of the particular sequence c^. However, this is not the case for the operator Sy (t). We have
Now if we example Cp = n!, we obtain
31 8y (t) x" = (X + y)", and if Cn = 1, we obtain
n + l ..n+l n X -y £ ,(t)x" = x-y
Proposition 4.1. Let U be a linear operator on P. There exists a series f(t) in F such that Up(x) = f(t) p(x) for all p(x) in P if and only if U commutes with the operators t, that is Utp(x) = tUp(x) for all p(x) in P.
The last equality follows from the fact (easily proved by induction) that deg Ux" < n and from Equation (3.4).
Corollary 1, a linear operator on P has the form f(t) in F if and only if it commutes with any delta operator.
Corollary 2. a linear operator on P has the form f(t) in F if and only if it commutes with any evaluation operator 8y (t).
By a sequence pn(x) in P we shall always imply that deg pn(x) == n.
Theorem 5.1: Let f(t) be a delta series an let g(t) be an invertible series.
Then the identity
;g(t)f(t)s„(x))=cA, (5-1)
for all n, k > 0 determines a unique sequence Sn (x) in P.
We will say that the sequence Sn(x) is the Sheffer sequence for the pair
(g (t), f(t)), or Sn(x) is Sheffer for (g (t), f(t)). Notice that g (t) must be invertible
32 and f (t) must be a delta series. The case g (t) = t° calls for special attention.
The Sheffer sequence for (f, f (t)) will be called the associated sequence for f(t), and we say Sn (x) is associated to f (t) or f (t) is associated to s„ (x).
Theorem 5.2: (The Expansion theorem). Let Sn (x) be Sheffer for (g(t), f(t)).
Then for any h (t) in F
k=o c^.
Corollary 1. If pn (x) is the associated sequence for f (t), then for any h (t) in F
h(,)=|:M|PiMf(,)'. k=0 C^
The next results show how to expand an arbitrary polynomial as a linear combination of polynomials from a Sheffer sequence. They follow from
Theorem 5.2 by taking h(t) = Sy (t).
Corollary 2. Let s„ (x) be Sheffer for (g (t), f (t)). Then for any p (x) in P P«=iM)[«J!M,w k>0 C^,
Corollary 3. Let pn (x) be associated to f (t). Then for any p(x) in P
^ f (')' P(x) p(x)=Z ^ ^Pk(x)- k>0 C|^
It is our intention now to characterize Sheffer sequences in several ways.
We begin with the generating function.
Theorem 5.3: (Generating Function). The sequence s„ (x) is Sheffer for (g(t).
f(t))ifandonlyif
33 for all y in K.
Corollary 1. The sequence p„ (x) is associated to f (t) if and only if
e,(f(.))=S ^^^^
Equation (5.3) defines the sequence c~' s^ (x) as a so-called generalized
Appell sequence by Boas and Buck. Thus we see that the present theory applies to a rather broad class of polynomial sequences.
The generating function leads us to a representation for Sheffer sequence.
Theorem 5.4: The sequence Sn (x) is Sheffer for (g (t), f (t)) if and only if
;g(f(t)rf(t) x" Sn(x)=I x\ (5.4) k=0 C,,
Theorem 5.5: The sequence Sn (x) is Sheffer for (g (t), f (t)) if and only if g(t) Sn (x) is the associated sequence for f (t).
Theorem 5.6: A sequence pn (x) is the associated sequence for f (t) if and only if
(i) (t°|p„(x)) = Co6„,o,
(ii) f(t)p„(x) = -^p„_,(x) Cn-1
34 Theorem 5.7: A sequence $„ (x) is Sheffer for (g (t), f (t)) for some invertible g (t) if and only if
f(t)sn(x) = -^s„_,(x). (5.5) Cn-I
Theorem 5.8: (The Sheffer Identity). A sequence s„ (x) is Sheffer for the pair
(g (t), f (t)) for some g (t) if and only if
S (t)s„ (x)= X -^^ p, (y)s„., (x). (5.6) for all y in K where pn (x) is associated to f (t).
One may observe that for Cn = n ! the Sheffer identity is
Sn(x + y)=Z L Sk (x)Pn_,< (x) k=o v*^; when Sn (x) = pn (x) = x" we get the binomial formula.
An important property of Sheffer sequence is their performance with respect to multiplication in F.
Theorem 5.9: Let s„ (x) be Sheffer for the pair (g (t), f (t)) and let pn(x) be associated to f (t). Then for all h (t) and l(t) in F we have h(t)i(t)K(x)>=s -^ii-(h(t)K(x)>{i(t)K_,(x)> k=0 Ck C„_k
6. RECURRENCE FORMULAS
If fi is a linear operator on P, then the adjoint |j.* is the linear operator on
F defined by
35 (f(')h>),>. ^•f(t)=Z k=0 ^-k
Proposition 6.1. If deg fj (t) > j for j > 0 and
f(t) = Ia^f,(t), j=0 then
^*f(t)=Xaj^*f,(t). (6.1) j=0
Proposition 6.2. If JA is a linear operator on P, then
(^*f(t)|p(x)) = (f(t)|^p(x)) forallf(t)in/^andp(x)inP.
Recall that a derivation d on an algebra A is a linear operator on satisfying
a(ab) = (5a)b + aSb for all a, b in A.
Theorem 6.1: An operator 0 on P is the umbral shift for f (t) if a only if its adjoint 9* is a derivation on F satisfying Eq. (6.1) and
e* fCt)"^ = kf(t)''"' for all > 0.
Proposition 6.3: If f(t) and g(t) are delta series, then
6;=(e;g(t))9;
Theorem 6.2; If 0,-and 0p are umbral shifts, then
36 6f=0g°(6;f(t))""
From this theorem we obtain our first recurrence formula.
Theorem 6.3: If Pn(x) is associated to f (t), then
(6.2) (n + l)c„ where f (t) is the ordinary derivative of f(t) with respect to t and 9t : x"
(n + lK n+l
V "-n+l J
Proposition 6.4. Let 9f be the umbral shift for f (t). Then
e;h(t) = h(t)ef-efh(t) for all h (t) in F.
Proposition 6.5. Let Sn(x) be Sheffer for (g (t), f (t)). Then if 9f is the umbral shift for f(t).
(n + lK
Theorem 6.4: Let Sn(x) be the Sheffer sequence for (g (t), f (t)). Then
Sn.,(x) = ^ ^x). (6.3) (n + lK (t)Jf'(t)
Theorem 6.5: Let Sn(x) be the Sheffer for (g (t), f (t)). If
37 T = Of- V g(t) J f'(t)
g(t)Jtf(t)' then
TSn(x) = nSn(x). (6.4)
In other words, Sn(x) is an eigenfunction for T with eigenvalue n.
Lemma 1. Let h(t) be invertible, with leading coefficient equal to 1. Then the
equation
h(t)= it) g(t)
has a unique (up to multiplicative constant) solution given by
g(t)=exp(|h(t)dt).
Lemma 2, Let l(t) be a delta series, with leading coefficient equal to 1. Then
the equation
f'(0
has a unique (up to multiplicative constant) solution given by
38 -1 ^ I(t)^ -1 f(t):=texp I^±i_.
Theorem 6.6: Let T be a linear operator of the form
T = (e,-h(t))i(t)
= (xD-th(t))ltJ where h(t) is invertible and l(t) is a delta series both having leading coefficient equal to 1. then a solution to the equation
TSn(x) = nSn(x) is given by nth polynomial in the Sheffer sequence for the pair
-1 \ m -1 expM h(t)dt),texp dt
The result follows from the lemmas.
We shall now derive another set of recurrence formulas.
Theorem 6.7: Let Sn(x) be Sheffer for (g (t), f (t)). Suppose
39 f(.)=Z ^t', k=l KC|j
glf(t)j k=i c,
Then
ns ,(x)=I (a^e,-djs„_k(x) (6.5) k=l >-c,k. c„_*-n-k. and
ns nW=Z (b,xD-dJs„_,(x) [do=0] (6.6) k=0 C|, C„_|^
Theorem 6.7 has a sort of converse.
Theorem 6.8: Let rn(x) be a sequence of polynomials satisfying (6.5). Let
k=l KC^
and
th(t)=Z ^t^ k=0 C|.
40 Then there is a constant a for which arn(x) is Sheffer for the pair
(exp(jh(l(t))dl(t)),l(t)).
Lemma 3. Let h(t) be invertible. Then there is a unique (up on muhiplicative constant) solution to the equation
l(t) = f^l f'(t) V <• } given by
f(.)=.exp(|l(iHd/
Theorem 6.9: Let rn(x) be a sequence of polynomials satisfying (6.6). Let
1(0= E ^t^ k=0 C|, and
th(t)=Z ^t^ k=o c^.
Then there is a constant a for which arn(x) is Sheffer for the pair (g (t), f (t)) where
f(0=.exp(|M:ild.'
41 g(t) = exp({h(l(t))dl(t)).
7. TRANSFER FORMULAS
In this section we develop formulas for the direct computation of association sequences. Using these formulas and Theorem 5.5 we can compute
Sheffer sequences.
Theorem 7.1: (The transfer Formula). If Pn(x) is the associated sequence for f(t), then
p„(x)=f'(t)f^T" X" (7.1)
for all n > 0. As usual f (t) is the derivative of f(t) with respect to t.
Theorem 7.2: (The transfer Formula). If pn(x) is the associated sequence for f(t), then
p„(x)=-^ec. , -f(t)V^ " X"-' (7.2) nc„_i I t ; for all n> 1.
We can use the Transfer formula to relate any two associated sequences.
42 Corollary 1. Let Pn(x) be associated to f(t) and qn(x) be associated to g(t) f(t), where g(t) is any invertible series. Then
q„(x)=e, g(tr e^' p„(x),
where 0^' x""^' = x" and e:' 1 = 0. (n + lk
The Transfer Formula readily gives Lagrange's formula for the compositional inverse of a delta series. To see this we take y = 0 in the generating function of a Sheffer sequence,
= y ^o^k(o) jk PD k=0 C.
since Eg (f (t))= c^'. Now by the Transfer Formula
,-k-l Cos,(0)= t^ g-'(t)f'(t) [ t
'(.)r(.)(f -k-i
.k Thus the coefficient of — in g"' (f(t)) equals c^ times the coefficient of in
-k-l g-' (t)f'(t) ^f(t)iiii ^ . Other versions of Lagrange's formula a similarly V t ;
derived.
43 Proposition 7.1: Let Sn(x) be Sheffer for (g (t), f (t)). Let h (t) and be invertible.
Then the sequence
rn(x) = h(t)l(t)"sn(x) is Sheffer for
''*"^<"l.(.).(.),.-'(.)f(.) f(t)h(t)
is the associated sequence for 1 ' (t) f (t), and the proof is complete.
8. UMBRAL COMPOSITION AND TRANSFER OPERATORS
Let pn (x) be associated to f (t). The transfer operator for Pn (x) or f (t) is the Unear operator A.f on P defined by
^fx" = Pn(x).
We have
,. g(,,^|. MKM.v (8,) k=0 CK
We can characterize transfer operators by their adjoints.
44 Theorem 8.1: A linear operator X and P is the transfer operator for f (t) if and only if its adjoint X,* is an automorphism of F satisfying (6.1) and for which
A.*f(t) = t.
Theorem 8.2:
(a) A transfer operator maps associated sequences to associated
sequences.
(b) If X : pn(x) -> qn(x) is a linear operator where pn(x) is
associated f(t) and qn(x) is associated to g(t), then A* g(t) =
f(t).
(c) If PnW and qn(x) are associated sequences and Xpn(x) = qn(x)
then A, is a transfer operator.
Theorem 8.3: Let pn(x) be associated to f(t) and qn(x) be associated to g(t).
Then qn(p(x)) is associated to g(f(t)).
Theorem 8.4: Let Sn(x) be Sheffer for (g(t), f(t)) and let r„(x) be Sheffer for
(h(t), l(t)). Then rn(s(x)) is Sheffer for the pair
(g(t) h(f(t)), l(f(t))).
Suppose Sn(x) and rn(x) are two sequence of polynomials related by
45 '•n(^)=Z an,kSw(x). k=0
The connection-constants problem is to determine the constants an, k- In case
Sn(x) and rn(x) are Sheffer sequences we can give a solution to this problem.
Theorem 8.5: Let Sn(x) be Sheffer for (g(t), f(t)) and let rn(x) be Sheffer for
(h(t), l(t)). Suppose
hM=t^nM\i^)- (8.2) k=0
Then the sequence
n
tn(x)=Z an.k>^k k=0
is the Sheffer sequence for the pair
Corollary 1. If Pn(x) is associated to f(t) and qn(x) is associated l(t) and
n
k=0
then t„(x)=^J]^Q a^ 1^ x^ is associated to l(f(t)).
46 9. EXAMPLES: GEGENBAUER, CHEBYSHEV AND
OTHERS
In this section we study the delta series
f(t)=-^^-'
This will lead us to the Gegenbauer and Chebyshev polynomials. Associated
Sequence
We begin by computing
f(t)= i+vr-' 7
r(t) -'' \ + x'
f'(t)= vr7)(i+A/r^) _ f(t) tVr^
We shall denote the associated sequence for f(t) by Pn(x). The generating function for pn(x) is
-2t^^|. P,(y) i + t'; ^0 c^
The conjugate representation for pn(x) is
47 ( -2t V
Pn(x)=Z k=0
Now
r-k^ f -2' 1 (t^J^^|x") j=0 J;
k '-k^ -(-2)^1 '^n "n,2j + k j=0 V J J
^2j-k^ (-2)"-^^ c, if k=n-2j . J . 0 if k ?in-2j and so
^J c f^j-n^ Pn(x)=Z (-2x) n-2j J=0 ^n-2j V J )
Let us give some consequences of the Expansion theorem. We have
..f;(!>Mf(,)V k=0 CK
But
t"|p,(x)) = 0 if n + k odd
48 and
r-n^ Pn.2.(x))-(-2)" 'n+2] V J ) and so
^-nV _t ^ n+2j t"=Z (-2)" j=0 J ) vl + VTVl-t^^ J or
(_^\ . _, ^-^ (i+VT^)r=i(-2)" j=0 \ J i+VT^ t^;
One can easily deduce (see Rainville [161]).
a(a + 2k-}X^ • z^ (9.1) (i-vr^r^i^^2a+Z^K j^. k=0 for all a. Hence
f(ty = X (-0' k(k4-2j-lV, ^,^^, (9.2) j=0 2^^^J j !
Thus we have
-'-Y. k=o CK
49 where by (9.2)
(f(t)'|x") = 0 if n-k odd and
n-2j (-l)"(n-2j)(n-l) f(t)' x") = j-i 2"j! and so
j=o ^ j! c'n-2. j
Now we turn to some recurrence formulas. From the fact that
^c„^ f(OPn(>^)= -^ p„_,(x)wehave V'-n-l J vr7p„(x)=(tf(t)+i)p„(x) (9.3) = -^tp„.,(x)+p„(x) c'n-. 1 and
•K^ p„(x)=(l + Vrv)p„(x)-p,(x)
=^(i+vn?')f(t)p„,(x)-p,(x) (9.4) 'n+l
-tp„„(x)-p„(x). 'n+l
50 Equating the two expressions gives the recurrence
^tp„„(x)+-^tp„,(x)+2p„(x)=0 (9.5) Cn+l ^n-1
Notice that (9.5) holds for any Sheffer sequence using f(t) as its delta series.
We may obtain another recurrence from (6.4). We observe first that
tf'(t)
and so
np,(x)=xDVi^p„(x)
and using (9.3) gives
(xD-n)p„(x)+^xDtp„_,(x) = 0. (9.6)
Also, Equation (6.6) becomes
"2] (xD-n)p,(x)=2xDt -^^^(-l)' p^.^M I<=1 ^2k''n-2k
Of course, Equations (9.5) and (9.6) may be used to derive other recurrences.
Now let us turn our attention to a specific form for Cn, namely.
51 c„ = " f-^.^
\ n y where -X is not a non-negative integer. Let us collect some preliminary results.
We have
Cn _ n c„_, -A,-n + l and
tx' .1-1 >.-n + l
One can easily verify that
{ = -{1 + xD)'' D. (9.7)
Also.
e, x" = - (?. + n) Xn+ l
and so
Bt - - X (X + xD).
Next we have
52 f'X^ ,w=z k k k=0 vky
= (l + yt)-k
The generating function now becomes
^-r \-^r-t :(y)t^ \ \+\' k=0 .k, or
^-^^ (l + t^ni-2yt4-t^)-^=X Pk(y)t^ k=0 V";
The conjugate representation yields
^ -X ^ ^2j-n^
(-2x) n-2.| Pn(x)=Z /^lA j=0 X,
V n y
Referring to (9.7) we see that (9.5) becomes
f-X\ f-l^ ^-x^ DPn.l(x) + DPn-lW-2 (X + xD)p„(x) = 0 (9.8) vn + ly vn-ly V n y which holds for any Sheffer sequence using f(t) as its delta series. Equation
(9.6) becomes (using (xD)^ = xD + x^ D^)
53 (x^D^+(?L-n + l)xD-?Ln)p„(x)+—-—xDVn-i(x)=0 X + n — \
Sheffer Sequence: Gegenbauer Case
Let Sn(x) be Sheffer for the pair (g(t), f(t)) where
f 2 ^'" g(t) = iW 1-t^
It is easy to see that
8{f(t))=(l + t^f- and so the generating function is
!-x\ (i+.'r"(i-2y<+''r=i SL.(y)i. ' k=0 vk,
f-l^ Thus when XQ = k, the polynomials S|.(x) are the Gegenbauer Vky polynomials.
To obtain the conjugate representation we notice that
-ji^f(,)'=(-2)^M,.,^r-
and a simple computation yields
54 f -I'
J 2j s„(x)=Z (-2x)"- j=0 f-X\
V n J Aat^9• ^^•.:'iJ 7 ,>->^*^
To apply the Expansion Theorem we observe that L{\- '0' g(t)fw=2^»(-t)^(i+vr^f"'' and from (9.1) we obtain
g(.)f(t)' =1; (-')H^.-M')(>.„^k^2j-i),., ^^,,^
j=0 2''+^jj!
Hence
^ -X ^
yn-2ky (-l)"(^o+n-2k)(A.o+n-lX., -"-Y. 'n-2k (>^)- k=0 2"k!
V n y
We may use the techniques of Section 8 to relate Sheffer sequences for different choices of A,o. Suppose Sn(x) is Sheffer for (go(t), f(t)) and suppose rn(x) is Sheffer for (gi(t), f(t)) where
go(t) = i+VT •t^ and
Vi g,(t) = i+VT^ t^J
55 If
II rn(x)=Z a„,kS,{x), k=0
II then by Theorem 8.5, t„ (x)= ^ a„ ^. x'' is ShefFer for the pair k=0
A Uolf(t)j which is the pair
Thus
t.{x) = (l4.,>)'--*"x"
'XQ -X, Z e'x" k=0 \ '^ J
( -I \ I] f A.Q -A,, 1 yn-2ky ,n-2k =0 f-y^'] V "^ y
V n y and finally
^ -A. ^ ^n-2kj X,Q -A., 'n-2k (x). k=0 ^-X^ V "^ /
V " ;
We now come to the recurrence formulas. First we need
56 g'(t) . y g(t) vr7(i+>/]r^)
= -Xotf (t) and
f'(t)
Then (6.4) gives
ns,(x) = xDVl^s„(x)4-^Xo ts„_, (x). 'n-l
Since (9.3) holds for Sp (x) we obtain
(n-xD)(X + xD)s„(x)-—^(Xo+xD)Ds,., (x) = 0. A + n-1
In case X, = XQ, for the Gegenbauer polynomials this becomes
xDSn(x)+ " Ds„_, (x)-ns„(x) = 0. (9.9) A. + n-l
We remark that Equations (9.8) and (9.9) are independent and can be used to derive all of the recurrences for Gegenbauer polynomials appearing in
Rainville [161] including the second-order differential equation.
Finally, Equation (6.6) gives
^-l^ 2I f-X^f -I ^ (n-xD>„(x)=2(xD + ^o)X (-I)'s.,,(x). V n J k=0 ,2k. \^n-2ky
Sheffer Sequences: Chebyshev Case
If?. = 1, then
57 and
t*,x" — X,,1- 1 and so
tp (X) = (-X)-' p (X).
The recurrence (9.5) becomes
2xp„(x) + p„„(x) + p„_,(x) = 0. (9.10)
The Sheffer seque4nce Tn(x) for the pair (g(t), f(t)) where
is related to the Chebyshev polynomials of the first kind. In fact we have
l + t_^ g(f(0)= 1-t^ and so the generating function for Sn(x) is
(l-t^)(l-2yt + t^)-'=X(-lf T,(x)t^ k=0
From T„(x)=g-'(l)p„(x) we get T„(X)=A/I^P„(X) and (9.3) gives
T„(x)=x-'p„_,(x)+p„(x) (9.11) and (9.4) gives
T„(x)=-x-'p,,(x)-Pn(x). (9.12)
The conjugate representation for Pn(x)
58 2j r2j-n^ Pn(x)=I (-2x)'n-2 j j=0 J ;
then gives direct formulas for Tn(x).
The Sheffer sequence Un(x) for the pair (g(t), f(t)) where ,(,)=^zivrz=_2iw
is related to the Chebyshev polynomials of the second kind. We have m-l + V and so
(l^2yt + .0"=i(-l)'U,(x).' k=0
Now
U„(x) = g-'(t)p,(x)
f(t)^ Pn(x)
^f(t)^ f(t)Pn.,(x) V I y
tPn+l(x)
1 x"'Pn.l(x) and so we see that the Chebyshev polynomials of the second kind are intimately related to the associated sequence Pn(x).
59 Combining (9.11) and (9.12) with (9.13) gives equations connecting
Tn(x) and Un(x),
Tn(x) = -2U„_2(x)-2xUn-,(x),
Tn(x)-2Un(x) + 2xUn_,(x).
And hence
Tn(x) = Un(x)-Un.2(x).
We conclude with the connection - constants problem
Un(x) = ia„,T,{x). k=0
n From Theorem 8.5 we see that t^(x)= J] a„ ^| x^ is Sheffer for (1 - t, t) and k=0
so
tn(x) = (l-tVx"
= Z (-ly t^^ X"
L2J = Z(-iyx-^ J=0
Thus an, n-2j = ("1)' and a^. n-2j+i = 0 so
U„(x)=Z(-iyT,^,^(x). J=0
60 10. EXAMPLES: JACOBI AND OTHERS
In this section we study the delta series
l + t-Vl + 2t f(t) =
l + t + Vl + 2t
Associated Sequence
First we have
21 f(t): {!-.)= •
f'(') = ^t^=f(<).
tVl + 2t
We shall denote the associated sequence for f(t) by Pn(x). The generating function is ^ 2t ^ ^y Pjjylj.
k=0 >-k
The conjugate representation for pn(x) is.
2^t^(l-tr PnW-Z k=0
Now
f 201k. \ 2^^(l-tP|x") = X 2''(-lWtJ^''|x" j=0 V J )
^-2k^ k/ i\n-k vn-k; '(-0
And so
61 P„(x) = I k=0
Our first recurrence comes from the equations
Vl^^ P„(x) = (l + t + ViT2^)p„{x)-(l + t)p„(x)
= _£j^(l + t + Vr72t)f(t)p„„(x)-(l + t)p„(x)
= -^tp„,,(x)-(l + t)p„(x) (10.1) c_, and
Vu2lp„(x) = (l + t-tf(t))p„(x)
= (l + t)p„(x) ^tp„_, (10.2) c'n-„ 1
From these equations we obtain
^tp„„(x) + -^tp„_,(x)-2(] + t)p„(x) = 0 (10.3) 'n-l
Which holds for all Sheffer sequences using the delta series f(t).
We obtain another recurrence from (6.4) by noticing that
f(t) = Vl + 2t tf'(t) and so
nPnl„(x)=xDVlT2x t p„(x).
Using (10.2) we obtain
^xDtp„„(x)-xD(l + t)p„(x)-np„(x) = 0. (10.4)
62 Sheffcr Sequence: Jacobi Case
We shall take
in) (1 + oc)' c„ = (l + a + p)r' r(2 + a + p) n)
2-^"(l + aJ(" ) (1 + a + pf"'
Since
4(a + n) c„_, (a4-p + 2n-!)(a + p + 2n)
One can readily check that
t = 4(l + a + P + 2xD)"'(2 + a + p + 2xD) ' x"'(a + xD). (10.5)
Also
(l+a + p)yM' f(2 + a + p)A^) ,{t)=Z y^t^ (Ua)'M
1+a+p 2+a+p 2F, 2 ; yt l + a;
We denote by Jn(x) the Sheffer sequence for he pair (g(t), f(t)) where
Y+a+P g(t) = l + Vl + 2t
Then since
g(f(t))=(i-tr-\
63 the generating function for Jn(x) is
1+a+P 2+a+3 -l-a-() ; 2yt (1-t) 2F, l.cc; '0-0'
(l + a + p)rrM (2 + a^P) (k)
,(k) J.(y)t^ k=0 (Ua)'
The classical Jacobi polynomials a denoted by Pf°'''(x) (see Rainville [161]).
j(x)J';"\f) c„p'°nx^i) (1 + a) '
The conjugate representation for Jn(x) is obtained from
g(f(t)rf(trx^l< \ =(2^tni-t/ik^kA »\-l-a-P-2) k
-l-a-p-2k z (-1)^2' (t^^Mx" j=0 V J
-i-a-p-2k (-ir2^c„ n-k
^a + P + n + k^ 2^c„ n + k
And gives
a+P+n+k ^(2x)^ k=0 n-k
n '^a + P + n + k^ (a + nLk 2n-k .,k •2^""'x =s n-k (a + P + 2nL. 2k k=0
64 We wish to express x" as a linear combination of Jk(x). From Corollary 2
to be Expansion Theorem (Theorem 5.2) we have
^"-Z ^ ^ J. (xX )
Now by (9.1) we have
(i+t+vrr2t)"' =2^(i+vmt)"2 H
^(2^ + 2j-l)^_, 2M+J-" j! (-ty j=0
and so
g(t)f(t)^=2'^"^^^(i+t+vrT2iy'"" p-k
^y . .,(l + a + P + k)(l + 2a + 2p + 2k + 24, ^,^,
J=0 ^ J •
and
^,.._(-ir(]+a + p-fk)(l + 2a + 2p + 2nU.. ^ g(t)f(tr ' 2"-'(n-k)!
Thus
y (-irO + a + P+k)(l + 2a + 2p-f2nt,_, c^^ ,, h 2"-'(n-k)! c, '^ -*
(l + a + p + k) (l + 2a + 2p + 2n)„.^_| (a + n)„_
k=0 (n-k)!(a + p + 2nL 2k
x(-l)''-^ 2"^^^^^' J,(x).
We conclude our discussion of the polynomials Jn(x) with some recurrence formulas. Equation (10.3) holds for Jn(x) where t is given Equation
65 (10.5). Let us derive Equation (6.6) of Theorem 6.7. From f(t)=2t(l-t)"' we obtain rmr f(,)..- V t J 1-t = • + 21 t^
and so
bo = Co,
bk = 2Ck.
Also, fromg(f(t))-(l _ Y+a+P we obtain
t|g(f(t))|' -(l + a + p) ^ g(f(t)) " ^ ^ 1-t
= - (l + a-t -P)Zt^ k=l
and
do = 0,
dk = -(l+ a + p)Ck.
Therefore Equation (6.6) becomes
(n-xD)j„(x)=X ^^(2c,xD + (l + a + p)cJj„_,(x) k=l ^y ^n-k
(l + a + P + 2xD)X -^J„.,(x). k=l '^n-k
66 11. EXAMPLES: THE q-CASE
In this section we shall briefly discuss the q-theory. We take
0-q)('-qO (l-q") c. = (l-q)"
Then
c„ _l-q" c.-, l-q and so
l-q
.x"-(qx)" x-qx and so
tp(x)=pWzPkO (11.1) x-qx
The q-binomial coefficient is
^n^
vky
(l-q)-;--(l-q"), (l-q) (l-q^)(l-q)...... (l-q"")
Thus we have
^n^ ^>(t)x"=I y" x"-" v"^; From (3.5) and the equations
67 5. t =7 xT* '
C-q)
a.f(t) = lW^Ifat) (11.2)
l-qt
yv^ y we obtaiyey(tn ) = ^(t)-Ey(qt-qt O or
B,(qt) = (l-(l-q)yt)s,(t) (11.3)
Sheffer sequences for the delta series f(t) = t satisfy
ts„W--p-s„_,(x) 1-q and in view of (11.1) we get
Sn(x)-Sn(qx) = (l-q")xSn_,(x). (11.4)
We define the sequence [x]a,n by
Ma,o= 1 ,
Ma,n=(x-a)(x-qa) (x-q""' a) and write [x]i „ as [x]n.
Then using (11.1) it is straightforward to verify that
1 — n " l-q
C„-i and so [x]a, n is Sheffer for the delta series f(t) ^ t. Therefore (11.4) gives
68 [X]a,„- [qx]a,n = (1 "Q") X [x]a,n-i
Since
(B.(t)|[xL) = [aL
= 5„,o , the sequence [x]a^ „ is Sheffer for the pair (63(1), t). From Theorem 5.5 and
(11.5) we obtain
Z ^ t^ [xL,„ k=o C^.
^n^ -I [xl,„-k vk;
The generating function for [x]a, „ is
ea(0 k=o (1-q) (l-q j
Letting y = 0 and noticing that [o]^ ^ = (- a)'' q^^"^ gives
.2j/(-at) -.\k' ^Mto (l-q) "(l-q^)^
Since
1[xL> = [oL
and using Theorem 4.1 we get
69 (.'|[xL)=:^(.1[4..-.> 'n-k
n-k = ^(-a)'n- k 'n-k
Thus be Corollary 2 of the Expansion Theorem
n\ n-k (-a) n-k q-^I 2 j X.. k k=0 v2y
Replacing x by ei(t) gives the formula
^n^ (B,(t)-1) (s,(t)-q"-')=Z n \^J
Applying this to a polynomial p(x) gives a formula which appears frequently in
the literature in the somewhat confusing form
n-k^l ^n> A"p(x)=Z (-ir q^^^p(x-fk). v2y
This same corollary also gives [.L.tfeM^kw k=0
/„A =z L^Jb,n-ki^Ja,k k=0 12A
70 This is actually the q-Vandermonde convolution formula in disguise. To this
notice that
[ax]aj, = a''[x]k
and so (11.6) gives, with x replaced by ax and b = 1.
[ax]„=S ^"^ [aL W- (11.6) k=0 vky
We wish to make the substitutions
x = q.
a = q , ax = q'^-".
Now if i [qJ], = (qJ-I)(qJ-q) (c^-q^-') ^j - • +1 = q '^'^ -'-'(q^-DlqJ-'-l) (q^" -1) = q^ (q^"-i) (q-1) (q-iy s (q-ir S-. = q^^^(q-iy ^ and if i > j, then [q"], = 0. Using this in (11.7) gives n-k 1 'n qkm ^1 2 n-k (q-1) C|+m-n k=0 C|^ C^_|, 71 ,k C, q^^Mq-O' 'm-n+k 'l-k rn-k^ (^ (r.\ which becomes, in view of + = -k(n-k), V 2 J v2y v2y O + m"^ nw m ^ Iq^k(m-n+k ) (11.8) V n y k=0 vk; n-kj Next we touch briefly on the q-Bemoulli polynomials. We define the q- integral by jx"=-C^„ ^n., -n+l 1-q ^n. 1-q n+l and I-' 1-q n+l -I The q-BernoulH polynomials have the generating function e,(t)-l k=o (1-q) (l-q ) That is, Bn(x) is Sheffer for the pair By Theorem 5.5 we have ^B„(x)=x" 72 But ,(<)-l n+l ^^^hJ^H tx 'n+l t = ^(E,(t)-l|x- 'n+l ^n in+l 'n+l = C X . x" and so 's,(t)-l p(x))= I xp(x) . In particular, Sn,0=JxB,(x) Corollary 2 of the Expansion Theorem is the q-Euler-MacLaurin Expansion P(X)\B,(X) k=0 We can also connect the two sequences Bn(x) and [x]n, Bn(x)=2. ^ ^L'^Jk k=0 /^r,^ z B„(l)[xl k=0 UA 73 and ,(t)-i M^) W=E W. k=0 Ck rn^ = 1 ,(t)-l Wn-k)B,(x) k=0 IkJc rn^ I Bk(x) Jx[xL k=0 vky We now turn to the q-Leibniz formula. First we need a lemma. Lemma. For any series f(t) and polynomial p(x), (a:f(t)|p(q^x)) = q-^"(5:f(q^t)|p(x)) The q-Leibniz formula is rr.\ a:(f(t)g(.))=I q-^("-^'aJf(t)arg(qt). k=0 vky The proof consists of the following calculations, in which we use the q-Vandermonde convolution (11.8) and Theorem 5.9, = ((f(t)g(t))|x"") '^n + m'' I (f(t)|x^)(g(t)|x-- j=0 J > n+m n+m ^n^ f m ^ q^(-^^^Vf(t)|x^)(g(t)|x- I Z vJ-ky j=0 k=0 v'^y 74 n+m /'n( n ^ n+nj..m« (I m \ , w . .1 \ / . xi = 1 Z . q^^'"-^^''\f(t)|x^){g(t)|x—^ k=o i^k;^ j=o VJ- rn\ rm^ =1 . I qMm-j)^f(^)|xj.ic^^g(t)|x—j- ^=« Vk>',J=« VV... Jy , ^n^ ^m^ = z S q^("^-j)(a;f(t)|x^)(ar'g(t)|x"^"^ k=0 vk, V j; ^«N ^m^ = 1 Z ajf(t)|x^)(arg(t)(q^xr k=0 vk; j=0 vj; ^n^ ^m^ = Z Z aff(t)|xnq-("-Varg(qH) , m-j k=0 vky j=0 u; /'„\ = z q-k(n-k)/5k f(^) 3„-. g(qk^) k=0 vky n = z q-^("-^)5jf(t)5rg(q4) k=0 kj. 75 CHAPTER - /// EXTENSION OF UMBRAL CALCULUS CHAPTER-III EXTENSION OF UMBRAL CALCULUS 1. INTRODUCTION The present chapter deals with the extension of Umbral calculus to more general class of sequences. In the previous chapter we studied a certain class of polynomials sequences called Sheffer sequences, by studying the dual vector space P* to the algebra of polynomials P. First we made P* into an algebra - the algebra of formal power series in t. Sheffer sequences were defined as those sequences in P which are orthogonal to some geometric sequence g(t) f(t) in P* (g(t)f(tf|s,(x)) = cA, . The degree requirements deg g(t) = 0 and deg f(t) = 1 were placed in order to ensure the existence and uniqueness of the sequences Sn(x). Now we extend the theory to a more general class of sequences in P* than the class of geometric sequences. This will bring some important new polynomial sequences into the purview of the umbral calculus. We shall call elements of this new class decentralized geometrical sequences. Recall that if Cn = n ! for all then the evaluation functional 8y(t) is the exponential series e^\ (e>^|p(x)) = p(y). 76 Now if yo, yi,... is a sequence of independent transcendentals (variable) then one decentralization of the geometric seque4nce t'' is the sequence. The intuitive reason for this is that .^=^-'—r—r~ -^ l5,( A&, No V' and \f<, <_ ^"^>^ (e^^'t^|p(x)) = p(^)(yj. — In words, one decentralization of the sequence "kth derivative evaluated at t^ is the sequence "kth derivative evaluated at y^." Now the seque4nce x" in P is orthogonal to the sequence t"^, x") = n!5„,, Thus, one decentralization of the sequence x" is the sequence Gn(X) satisfying (e^^V|G„(x)) = n!5„,, or rather. The polynomials G„{x) are known as the Goncarov (or Gontscharoff) polynomials and are of considerable important in the theory of interpolation. Thus, the umbral calculus may be extended to include the Goncarov polynomials. Another important polynomial sequence which we shall incorporate in this way is the sequence Sn(x) = (x-yo)(x-y,) (x-yn.,). 77 Up to now we have been using the algebra of formal power series in a single variable to study the dual vector space P* and hence also the algebra P. We have come to a point in the development of the umbral calculus where to pays to reverse this point of view. Accordingly, we devote a section of this super to the study of the algebra of formal power series in a single variable by thinking of formal power series as linear functionals on P. 2. DECENTRALIZED GEOMETRIC SEQUENCES To fix our terminology we shall let P be the algebra of polynomials in x and F be the algebra of formal power series in t, both over the base field K of characteristic zero. We let Cp be a sequence of nonzero constants. The algebra F represents the set of all linear functionals of P as well as certain of linear operators on P - this by means of x") = c,5„,. and \ f c„ t^x^ = x"-*^ v^n-k y for details one is referred to chapter - II. Lemma 2.L If def fk(t) = k and (f^(t)|p(x)) = 0 for all k > 0, then p(x) - 0. Similarly, if deg pk(x) = k and (f(t)|pk(x)) = 0 for all k > 0, the f(t) = 0. Suppose yo, yby2 and zo, z,,Z2 78 form a set of independent transcendentals (i.e., variables). Let k be the quotient field of the algebra of all formal power series in these transcendentals over a base field C. We need not concern ourselves with a prescribe description of K. Our only concern is that K is a field. It is critical to the theory to assume that the constants Cn lie in C and, hence, are independent of the transcendentals. For any integer k we define the function S*^ on K which shifts the subscripts of the transcendentals. Thus, ^ yj ^ yj+k' s Zj=Zj+k. Notice S"^ yj is not defined if k + j < 0 and we shall be careful to and such a contingency. Now suppose f(t) is a series in F. Then k < 0 we shall write S'^f(t) as f(t;k). For uniformity we may write f(t) as f(t:0). Then put it in words, f(l) may involve some transcendentals in its coefficient. Then f(t;k) is obtained from f(t) by adding k to the subscripts of these transcendentals. We adopt a similar notation S'^p(x) = p(x;k) from polynomials in P. h,(t) = g(t;k) f(t;0) f(t; k - 1), ho(t) = g(t;0), where g(t) and f(t) are in F. Whenever g(t) and f(t) do not involve the transcendentals the sequence hk(t) reduces to a geometric sequence in F. We shall have frequent occasion to use the fact that S^(f(t)|p(x)> = (s^f(t)|s^p(x)) (2.1) and Slf(t)p(x)l = [s^f(t)][s^p(x)] (2.2) 79 whenever all expressions are defined. In particular, if (f(t)|p(x)) is dependent of the transcendentals, then (f (t)|p(x)) = (f (t;k)|p(x;k)) for all k > 0. 3. DECENTRALIZED SHEFFER SEQUENCES As usual by a sequence pn(x) in P we imply that deg pn(x) = n. Recall that a delta series f(t) in F is a series satisfying deg f(t) = 1 and an invertible series g(t) is a series satisfying deg g(t) = 0. Theorem 3.1. Let f(t) be a delta series and let g(t) be invertible. Then the identity {g(t;k)f(t;0) f(t;k-l)|s„(x)> = c„6„,, (3.1) for all n , k > 0 determines a unique sequence Sn(x) in P. Theorem 3.2: (The expansion theorem). Let Sn(x) be decentralized Sheffer for (g(t), f(t)). Then for any h(t) in F, h(t)=|; <^^M^g(t;k)f(t;0) f(t;k-l). k=0 C|^ Corollary 3. Let Sn(x) be decentralized Sheffer for (g(t), f(t)). Then for any p(x) in P, p{x)=iMiiMH°i=I We now give an operator characterization of decentralized Sheffer sequence. 80 Theorem 3.3: The sequence Sn(x) is decentralized Sheffer for (g(t), f(t)) if and only if (1) (g(t)|s„(x)) = c„ 5„,o (2) f(t)s„(x) = |-^ls„_,(x;l) We now come to the decentralized Sheffer identity. Theorem 3.4: A sequence Sn(x) is decentralized Sheffer for (g(t), f(t)) for some delta series f(t) if and only if deg g(t) = 0 and Sv(t)s„(x)=I -^^s,(y)g(t;k)s„_,(x;k) (3.2.) for all y in K and for all n > 0. 4. COMPUTATION OF DECENTRALIZED SHEFFER SEQUENCE In order to derive formulas for the computation of decentralized Sheffer sequences we are required to make some slight extensions in the theory. Let f(t) be a delta series. Then f(t) has no multiplicative inverse in F. Though of as a formal Laurent series, however, f(t) has a multiplicative in denoted by r'(t) and satisfying rrffW f-'(t)=t- V t J 81 We shall require the use of the linear operator f' (t) on P define in extending the definition r ^ \ .n-k t''x" = X V'-n-k / for all k > 0 to include r „ ^ t-" x" = n+l v^n+1 y To be absolutely clear, if f ' (t) = ^ ^\.^^, then k=-l f-'(t)x" = Sa,t^x" k=-l f ^ \ n-k k=-l V^n-ky We shall also require the use of the linear functional T (t). In this case we set t"' x") = 0 for all n > 0. Thus we have r'(t)lx")= Za,.' Vk=0 where we have simply dropped the term involving t '. If g(t) is in F the expression (g(t)f '(t) x") is evaluated by first taking the formal product g(t) f'(t) before dropping any terms involving t -1 82 Theorem 4.1: Let f(t) be a delta series and let g(t) be any series in F. Then (g(t)f-(t)|x">=(g(t)|r'(t)x"). Theorem 4.2: Let f(t) be a delta series and let g(t) be any series in F. Then as linear operators we have g(t)o r'(t)=g(t)f-(t), where the left side is composition of operators and the right side is the product of formal Laurent series. Theorem 4.3: Let Sn(x) be the decentralized Sheffer sequence for (g(t), f(t)). Then ^ c„ ^ ax) = Jn°J. ° °Jn-n-Ii 1 . vCoSny where by Jo°J|° °K-\^ ^^ mean the composition of the operators JQOJIO oj^ I applied to the polynomial p(x) = 1. If n = 0, then JQ O o J^ I is the identity operator. Again gn is the constant terms is g(t; n). Also, since deg g(t; k) = 0. Theorem 4.1 gives (g(t;k)|j,x") = (g(t;k)|f-'(t;k)x"; g(t;k)f-'(t;k)|x")(g(t;k)|l> .Cog k J = (g(t;k)|f-'(t;k)x")-(g(t;k)f-'(t;k) X (4.1) 83 We may derive another formula for decentralized Sheffer sequences. Let f(t) be a delta series and let g(t) be invertible. We define the linear operators Hi onP by Ho = (g(t;0))-'t-' and for k > 0 (g(t;k-i)) H,= f-'(t;k-i). g(t;k) We denote the leading coefficient of the series H^ by h^. Theorem 4.4: Let Sn(x) be the decentralized Sheffer sequence for (g(t), f(t)). Then c. ax) = hnHooH.o oH,_,l, V^o y 5. THE CLASSICAL UMBRAL CALCULUS The term classical umbral calculus will refer to the case Cn = n! for all n > 0. In this case the operator t"^ is the ordinary kth derivative and the linear functional t'' is the kth derivative evaluated at 0. e1p(x)) = p(y). Therefore, the product of evaluation at y with evaluation at z is evaluation at y + z, 84 The operator e^ is translation by y. e^ p(x) - p (x + y). ^n^ The expression — becomes the familiar binomial coefficient and vky the decentralizer Sheffer identity is ^n^ Sn(x + y)=Z s^(y)g(t;k)s„_,,(x;k) k=0 ^k; The Goncarov Polynomials Let us take g(t) = e^«' and f(t) = t. Then the decentralized geometric sequence for (g(t), f(t)) is hk(t) = e^^'t\ where (h>(<)|p{x)) = pW(y,). The decentralized Sheffer sequence for (e^"',t) is the Goncarov sequence. We shall soon see that the nth Goncarov polynomial involves only the transcendental yo, , yn-i and we shall use the notation Gn(x; yo, , yn-i) for the nth Goncarov polynomial. From the definition we have Gr(y,;yo, ,yn-,)=n!5„. k • 85 Theorem 3.3 characterizes the Goncarov polynomials by Gn(yo;yo, ,yn-i) = Sn,o G'n(x;yo, , yn_i) = nGn-i(x ; y,, ,yn-i)- The decentralized Sheffer identity gives Gn(x + y;yo, , yn-i) = 1 Gk(y;yo' .Yk-i) G„_k(x + y,;y,, ,yj k=0 vky which is a new identity for Goncarov polynomials. As a application of the expansion theorem we expand the polynomials Gn_i(x;y,, ,y„_,) in terms ofGk(x;yo, ,yk_,). Corollary 1 of Theorem 3.2 gives xGn_, (x;yi, , yn-i) n (e^^'t' xGn.|(x;y,, ,y,^,y =1 Gk(x;yo, ,yn-i)- k=0 k! But recalling that (f (t)|xp(x)) = (f'(t)|xp(x)) we have e^'^' t'^ xG„_, (x;y,, ,y„_,)' = ((e-t^)' G„-, (x;y,, ,y„_,); = (y,e^^'t^+ke^^'t^-'|G„.,(x;y,, ,y„_,); = yk(n-l)k G„_,_, (y^; y^.i, ,yn-i) + k(n-lX_, G„_Jy^; y,, ,y„_,) 86 = yk(n-OkG„_,_k(yk; Yk^,, ,y„_,)+n!5„_ and so n-l (n-\\ xG„_, (x;y,, ,y„_,) = G„ (x;yo, ,yn-,)+Z Yk k=0 V k , xGn-1-k (yk;yk+i> 'yn-i)=Gn (x;yo. .yk-O- This result is due to N. Levinson, for use in obtaining a bound for which taker's constant. A formula for the Goncarov polynomials may be obtained from Theorem 4.3. In this case T' (t; k) == t~' and so J, x"=t-'x"n _ t-1 ^n - /„yie^^'t t.- 1 :(n + l)-'x"^'-(e^^' t-'x" 1 ..n+l (n + ir'x"^'-(n + l)-'y k = ^"^- From this follows the usual formula Gn(x;yo. .yn-i)=n! ^^ dt, ^' dtj ^"^[ dt. If the difference operator Ay is defined by Ay p(x) = p(x) - p(y) and if we suggestively write t"' as D~' we have J,x"=A^^oD-'x" and so J,'k. "= A'-'„v o D ' . Thus, G„(x;yo, ,y„_,) = n!A^ oD-'o OA,_,OD-'1 87 Theorem 4.4 gives a different, and apparently new, formula for the Goncarov polynomials. We have Ho=e-^"'t-' and for k > 1 H, =e(^^-'-^^^'t-'. Then G„(x;yo, ,yn-,) = n! e"^^' t"' oe^'"'-'")' t"' o oe*^"-"''-'^' t"'. If we denote the translation operator e^ by T^, then we obtain (with t~' written asD"^') G.(x;yo, ,ynJ=n! T.^„ OD- OT^,^,_^,^ OD"' O OT^,^^_^^ , OD- . A generalization of the Goncarov polynomials is obtained by taking g(t) = e^"' andf(t)tobe any series which is independent of the transcendentals. The decentralized Sheffer sequence for the pair (e^"', f (t)) satisfies {e^^'f(ty|s,(x)) = n!5,,. In other words, Sn(x) is the sequence of interpolation polynomials for the sequence of linear functional s^, (t) o f(t) . From Theorem 4.3 we have J,x"=f-'(t)x"-(e^^^f-'(t)x = Ay^ or'(D)x" and so s,(x) = n!A,,,,of^'(D)o oA^^, of-'(D)l. 88 from Theorem 4.4 we have the alternative formula ^n!^ VM y 6. THE NEWTONIAN UMBRAL CALCULUS The term Newtonian umbral calculus will refer to the case Cn= 1 for all n > 0. The operator t is multiplication by x" tx" = x"-'. where tl = 0. The evaluation functional is the geometric series s(t)=Z/t^- k=0 1-yt and so lp(x)) = p(y). 1-yt The operator e (t) = (l - yt) ' satisfies n-l _ n + l k „n-k ^ y 1 ~ yt \=0 k=0 x-y and so p(^)^xp(x)-yp(y) 1-yt x-y Thus, the decentralized Sheffer identity becomes 89 xs I"^'^"^^"^^^-=Zs,(y)g(t;k)v,(x;kI \ ) x-y n3 Let us compute the decentralized Sheffer sequence for ((l-Yot) ' ,s). From theorem 4.3 we obtain in the usual way s„(x;yo, ,yn-i) = K° ^ ° °^y,., ° ^^ > Where X is the operator multiplication by x. This sequence is the Newtonian analog of the Goncarov sequence. It satisfies (S.(t)|x-^s„(x;y„, ,y„J) = 5„,, And X"' s„(x;yo, ,y„^i) = s„_,(x;y,, ,y„.,) . where if course X "'' x-" = 0 if j - k < 0. Notice that the delta series t8 (t) = -, r satisfies >^^ (1-yt) , n ,_n ' .x-=-L.,...''•-/• 1-yt 1-yt x-y and so ^ t ^ |-^,p(x) = ^e^^ (6.1) (l-y)j (x-y) which is the first divided difference of p(x). The kth divided difference of a polynomial p(x) with respect to a sequence y^ is usually defined by a recurrence p[yo] ] p(yo) p[yo,y,]=%tp[yJ (yo-y,) 90 r ,, i_(p[yo yi-il-p[yp >ykD We are now in a position to describe these divided differences directly. We begin by observing that 1 '[yo] = < P(x) 1-yt and '^'"'''^ (yo-y.)lp(x)) P(x) ^O-yoOO-y.t) and (i-yot)(i-y,t) (i-y,t)(i-y,t). p[yo>ypy2]= P{x) yo-y2 P(x) ^(l-yot)(l-y,t)(l-y,t) The pattern is now clear. The decentralized geometric sequence h.(0-S,(t)tS„(t) ts^^_,(t)= (i-yoO (1-y.t) Has the property that u (,x_(h,(t)-S'h,(t)) (yo-yK.J and since (ho(t)jp(x)) = p[yj we deduce that pW) = p[yo> .yicl- ^(i-yoO O-y.t) Notice that hk(t) is the centraHzed geometric sequence for g(t) = (1 - yot)" 1 = S„(t)andf(t) = ^^--^ = te^„(t). Naturally we want next to determine the decentralized Sheffer sequence '(i-yotr,t^ for In order to use Theorem 4.4 we compute (1-yot) ) Ho = (l-yot)t"'=t-'-yo = X-yo, Where X = t"' is the operator multiplication by X. Also, for k > 0. H, -t -y|,t-X-y^. . i-yk-,t t 0-yotr,1 t .^ Thus the decentralized Sheffer sequence for IS , (i-yoO , ^n(^;yo' .yn-i)=(x-yo)(>i-yi) (x-yn-i)- Thus, from the definition of decentralized Sheffer sequence ^n[yo' ,yk] = 5n,k- Theorem 3.3 characterizes 7i„(x;yo, ,yn-i) ^s the unique polynomials sequence for which ^n(yo' .yn-i)=s„,o and 92 (7in(x;yo^ .yn-i)-"n(yQ;yo> ^y,^-^)) (x-yo) = ^n-i(x;yi, ,y„-i)- It is interesting to notice that f\ + k\ yj(t'^j|x") .(1-ytr j=0 J J ^n^ .n-k Vky i~(^,(')|D'x" t' 1 and so —- is — times the kth derivative evaluated at y. We may use (1-ytf k! this result to handle the case of confluent transcendentals. In particular suppose yo = yi = = yj for j < k. Then using (6.1) we have .k+l p[yo,yM ^Yk-i]^ P(x) .0-yot) (i-yk-it) .k-H P(x) \(i-yot) j+i (l-yj.,t) (l-y,_,t) S,,W|D'p[x,yH' ^yk-i]) J! It is worth taking special note of the case of two transcendentals. p(y)-p(z)_/ t P(x). y-z ,(l-yt)(l-zt) Now we would like to set y = z, which can be done in the right side and gives 93 p(x)U(s,(t)|Dp(x)) .(1-yt)^ 7. FORMAL POWER SERIES Up to now the umbral calculus has taken the point of view that the dual- vector space P* can be profitably studied via the algebra F of formal power series in a single variable. We shall now take the reverse point of view. Since any linear functional on P is a formal power series - so any formal power series is a linear functional. In fact, a formal power series is a linear functional in many ways - one for each choice of the sequence Cp. A more detailed study of the algebra of formal - power series is given in the IV chapter of this thesis. Let Cn and dn be sequences of nonzero constants. Let g(t) and h(t) be invertible series in F and let f(t) and l(t) be delta series. Consider the formal power series. u(t)=i; ^g(t;k)f(t;0) f(t;k-l) k=o c^ for constants a^ in K. Then u(t) has an expression of the form CO L, u(t)=Z ^h(t;k)l(t;0) l(t;k-l) for some constants b^. We wish to determine the constants b^ in terms of the constants a^. Since we are considering more than one sequence of nonzero constants, we have more than one action of F on P. We shall use the notation 94 {f(<)lp(>')>, to denote the action of the linear functional f(t) on p(x) derived from t'|x") = c„5,, . Similarly, we shall write (f(t)|p(x))^ for the action of f(t) on p(x) derived from x" \ = d„ 5„,( . The following lemma will be use: Lemma 7.1. If u(t) is in F, then u(t)|x")^ (u(.)|x") dn C„ Theorem 7.1: Let u(t) be the formal series u(t)=X ^g(t;k)f(t;0) f(t;k-l). (7.1) where a^ is in K. Then CO U u(t)=Z ^h(t;k)l(t;0) l(t;k-l) (7.2) k=o d^ where n (u(t)|x^) , , j=0 <-j where rn(x) is the decentralized Sheffer sequence for (h(t), l(t)) using the sequence of constants dn. That is, (h(t;k)l(t;0) l(t;k-l)|r„(x))^=d„6„,. Theorem 7.2: Let u(t) be the formal series 95 u(t)=Z ^g(t;k)f(t;0a ) f(t;k-l), k=0 C^ where a^ is in K. Then u(t)=Z ^ h(t;k)l(t;0) l(t;k-l), k=0 dk where K=t ^ (g(t;j)f(t;0) f(t;j-l)|r,(x)), j=0 Cj where rn(x) is the decentraUzed Sheffer sequence for (h(t), l(t)) Corollary 7.1, Let CO „ u(t)=Z ^t k k=0 C|, Then u(t)=Z ^h(t;k)l(t;0) l(t;k-l),. k=o d^ where b, = I^(t'kW), j=0 Cj where rn(x) is the decentralized Sheffer sequence for (h(t), l(t)). Let us give some examples. Example 1. Let u(t)=f: ^t k ^ ^ 6 k! We determine bn in 96 k=0 K1 referring to Corollary 7.1 we have €„= n ! and h(t) = e''»', l(t)=t and so rn(x) = Gn (x; yo, ,yn-i) is the Goncarov sequence. Thus, K=t ^(t^|Gn(x;yo, ,yn-.) 1=0 J! r..\ = 1 'j Gn-j{0;yj, ,y„_,) j=0 vjy and we may write 00 „ "O 1 rv\ y ^t^=y -L ajGn_j(0;yj, ,y„_,) e-^^'t' &o k! ,^0 k! j=0 u; Example 2. Let u(t)=Za,t^ k=0 We determine bn in u(t)=Ib, k=0 (l-Yot) (l-Ykt)' Referring to corollary 7.1 we have c„ = 1 and 97 h(t)=-^, l(t)=. ^ l-Yot l-Yot and so rn(x) = (x-yo) (x-Yn-i) Thus, b„=Zaj(t1(^-yo) (x-Yn-,))- j=0 If we write a (n, j) for the elementary symmetric function on yo, , Yn -1 of order j, then (x-Yo) (x-Y.J = Z (-lN(n;i)x"-' i=0 and so tl(x-yo) (x-y„j) = (-ira(n,n-j). Therefore, bn=I (-ira^a(n,n-j) j=0 and CC 00 Z (-ir''aja(n,n-j) k=0 k=0 j=o (l-YoO (l-YkO Example 3. Let u(t) = Z ^e-'t^ k=o K! 98 We determine bn in u(t)=Zb to ' (l-ZoO (1-Zkt) • Referring to Theorem 7.1 we have c„=n! g(t) = e^"', f(t) = t^ and d„=l h(t)=(l-Zot)-', 1(0 = 7-^. and so '•nW^lx-Zo) (x-z„.,) = 2: (-1)^ a(n,i)x"-'. (7.3) i=0 Therefore bn=Z ^^^(tl(x-Zo) (X-Z,,,) r^ J! We see that tl(x-zo) (x-z„J) =(-irJa(n,n-j) and 00 O u(t)|xO,=E77(^''=0 J! ^ J rj^ j-i I SJY; i=0 VJ Hence 99 n J (-0'..-jc n b.=II a^yf a(n,n-j). j=0 1=0 j! \h Example 4. Let u(t)=Z^ k=0 (1-yot) (i-y^t) • We determine bn in u(.)=i^e-t'. k=0 '^• Referring to Theorem 7.1 we have c„=l, g(t) = (l-yot)-', f(t): (l-Yot)^ and d„=n! h(t) = e'"', l(t)=t Therefore rn(x) = Gn (x; zo , ,Zn_,) and K=t (uW|x^) (tiG„(x;to, ,z„j) j=0 Now t'|Gn(x;Zo, ,z„.,)) =(n)jG,_3(0;z^, ,z„_,) and y(t)|x') =y a,/^ ^. ^,xA. 100 If we use the notation [yo, , yk]xj for the kth divided difference of x-', then (u(t)|x^)^=taJyo, ,yX ' i=0 and so nII JJ / V bn=Z E Hajbo' .yi]xjG„_j(o;zj, ,z„_,). j=0 i=0 CHAPTER -IV GENERALIZED UMBRAL CALCULUS CHAPTER-IV GENERALIZED UMBRAL CALCULUS 1. INTRODUCTION The present chapter deals with generalization of Umbral Calculus. In this chapter we discuss in some detail the cases of Hermite, Laguerre, Gegenbauer, and the q-case but only mentioning briefly the Chebyshev and Jacobi cases. The underlying theme of the second chapter is that the special polynomial sequences which we have termed Sheffer sequences posses a unified theory. From this theory many seemingly unrelated resuhs in the classical literature emerge as special cases of a general result. Moreover, the umbral calculus provides a cohesive approach to the study of further properties of these sequences, as well as bringing to light some new sequences which are strongly related to the important classical ones. In this chapter we replace the algebra P of polynomials in x by the field A of formal Laurent series in x of the form m p(x)= Z b^ x^ for any integer m. The algebra F of formal power series in t, which gave us our representation of linear functional and linear operators on P, is replaced by the field r of formal Laurent series in t of the form 102 f(t)=Za,t\ k=n for any integer n. Then F is used to represent all the continuous linear functionals and some continuous linear operators on A in much the same manner as in chapter - II. In fact, we still have x") = Cnd„,, and Cn-k but now n and k range over all integers. Each Sheffer sequence of polynomials k=0 gives rise to a sequence of Laurent series n Sn(x)= Z a„,kx', k=-co where n ranges over all integers and for which a^ ^^ =a^^. whenever n, k > 0. In some special cases, and only n < 0, the sequence s„(x) has been studied under the name factor sequence. Several simple examples have appeared in the classical literature but it was not until the last few years that the factor sequence counterparts of, say, the Hermit and Laguerre polynomials, have emerged. However, except for some brief work by S. Roman the full sequence s^ (x) has not been studied. 103 2. FORMAL LAURENT SERIES In this section we shall set down a few basic definitions concerning formal Laurent series. Let K be a field of characteristic zero. Let F be the field of formal Laurent series over K of the form f(t)=i;a,t\ (2.1) k=m where m is any integer. Addition and muUiplication in F are purely formal. The degree of f(t) is the smallest integer k for which a^ i^ 0. It is readily seen that deg f(t) g(t) = deg f(t) + deg g(t). We shall denote the multiplicative inverse of f(t) in F by r'(t) or l/f(t). It is clear that deg r'(t) = -deg f(t). We shall write r'(t)^asf(t)"^ Let gi^(t) be a sequence in F for which lim^^oD deg gk(t) = oo. Then for any sequence of constants a^ the sum 00 Z a,gk(t) k=m is a well defined element of F. In case deg gk(t) = k the sequence gi;(t) forms a pseudo basis for F. In other words, for any f(t) in F there exists a unique sequence of constants a^ and integer m for which f(t)=Sa,g,(t). k=m If deg g(t) = I and f(t) has the form (2.1), then the composition r(g(t))=i;a,g,(ty k=m 104 is a well defined element of P. A series f(t) has a compositional inverse, denoted by f (t), and satisfying f (f(t))= f (f (t))= t, if and only if deg f(t) - 1. We call any series f(t) with deg f(t) = 1 a delta series. It will be convenient to define a notion of convergence in F. The sequence fn(t) converges to 0 if limn-»ao deg fn(t) = oo. In case f(t) = 0 we set deg f(t) = 00. If fn(t) converges to 0 we write fn(t) -> 0. The sequence fn(t) converges to f(t) if fn(t) - f(t) -^ 0. A linear operator T on F is continuous if and only if Tfn(l) -> 0 whenever fn(t) -^ 0. In particular, a linear functional L is continuous if and only if Lfn(t) is eventually equal to 0 whenever fn(t) -> 0. In case deg fn(t) = n, where n ranges over all integers then a continuous linear operator T on F is uniquely defined by the values Tfn(t). Moreover, these values may be assigned arbitrarily provided Tfn(t) ^ 0 as n ^ oo. Let A be the field of formal Laurent series over K of the form m P(x)= I b^ x^ for any integer n. Addition and multiplication are formal and the degree of p(x) is the largest integer j for which bj ?^ 0. If pj(x) is a sequence in A for which limj_),„ deg pj(x) = - co and if bj is a sequence of constants, then m z b^ p;(x) J=-CC is a well defined element of A. 105 We shall say that the sequence pn(x) in A converges to 0 if limn -* a> deg Pn(x) = - 00. Here if p(x) = 0 we set deg p(x) = - oo. In this case we write Pn(x) -^ 0. The sequence pn(x) converges to p(x) if Pn(x) - p(x) -^ 0. A linear operator T on A is continuous if Tpn(x) -> 0 whenever Pn(x)->0. A linear functional L is therefore continuous if and only if Lpn(x) is eventually 0 whenever Pn(x) -> 0. If deg pn (x) = n for all integers n, then pn(x) is a pseudobasis for A and a continuous linear T on A is uniquely defined by the values Tpn(x). Moreover, these values may be assigned arbitrarily provided Tpn(x) —> 0 as n -> -co. 3. LINEAR FUNCTIONALS Let A* be the vector space of all continuous linear functionals on A. We use the notation (L|p(x)) for the action of L in A* on p(x) in A. Let Cn be a fixed sequence of nonzero constants. If f(t) in T has the form f(t)= Z a, t^ k=ni then we define the continuous linear functional f(t) on a' by (f(t)|x") = c,a„ (3.1) for all integers n, where an = 0 if n < m. In view of the remarks at the end of Section 2, since deg x" = n and Cn an —> 0 as n —> - oo we conclude that (3.1) defines a unique element of A*. 106 Notice that (3.1) gives for all integers n and k. We have used the same notation f(t) for a formal Laurent series and a continuous linear functional on A. No confusion should arise since f(t) = g(t) as formal Laurent series if and only if f(t) = f(t) as continuous linear fijnctionals. Any continuous linear functional L on A has the property that L x") -^ 0 as n -> - CO. In other words there exists an integer no, depending on L, for which n < UQ implies (L x") = 0. Now consider the series f,-W= I Sr^ > k=n„ C Then f.(t) x" = L for all integers n. Therefore, since both L and fL(t) are continuous we conclude that L = fi (t) as linear functionals. The upshot is that the map v|/: L -^ fL(t) is a vector space isomorphism from the vector space A* onto the field F. As is usual in the umbral calculus, we shall obscure this isomorphism and think of A* as being identical with f. Hence A* is a field. Let us give some basic some basic facts. Proposition 3.L If deg f(x) > deg p(x). then (f(t)|p(x)) = 0. 107 Proposition 3.2. Let deg fk(t) = k and deg Pj(x) = j. Then 11 \ Za,f,(t) \k=m J=-00 I Za,b3(f,(t)|p^(x)) k=m j=-oo IZa,b,{f,(t)|p^(x)). k=m j=m Proposition 3.3. For f(t) in T k=m C where m = deg f(t). Proposition 3.4. For f(t) and g(t) in F n-s ^ f(t)g(t)x" =Z -^:^f(t)x^ g(t)x- , k=m Cj. C^_^. where m = deg f(t) and s = deg g(t). Proposition 3.5. For p(x) is in A, then J=-co Cj where n = deg p(x). Proposition 3.6. If deg fk(t) = k and (f^. (t)|p(x)) = 0 for all integers k, then p(x) = 0. Proposition 3.7. If deg pn(x) = n and (f(t)|p„(x)) = 0 for all integers n, then f(t) = 0. 108 If y is a constant and m is an integer we define tiie evaluation series (or evaluation functional) of degree m by k=m ^k Then Kn.W|x") = y" if n>m, -0 if n>m. Proposition 3.8. Let p(x) and q(x) be in A. Then p(x) = q(x) if and only if 8y m (t) p(x) = 8y m (t) q(x) for all y in K and all integers m. 4. LINEAR OPERATORS If f(t) in r has the form f(t)=Ia,t^ k=m we define the continuous linear operator f(t) on F by CO „ n-m p f(t)n" = X ^a,x"-^=X ^a„.,x^ . (4.1) k=m ^n-V k=-oo C|; In particular t^x^^-^x"-" Cn-k for all integers n and k. Notice that we use juxtaposition for the action of a linear operator. 109 We use the same notation f(t) for a formal Laurent series, a linear functional and a linear operator. Again it is easy to see that f(t) = g(t) as formal Laurent series if and only if f(t) = g(t) as continuous linear operators. It is straightforward to verily that f(t)|g(t)p(x)|Hf(t)g(t)|p(x) =UWf(t)|pW = g(t)|f(t)p(x)| for all f(t) and g(t) in F and p(x) in A. Notice that by Proposition 3.3 Equation (4.1) becomes f{t)x"=I -^^(f(t)|x"-'')x'' . (4.2) Theorem 4.1: For any f(t) and g(t) in F we have (f{t)g(t)|p(x)) = (f(t)|g(t)p(x)> for all p(x) in A. It is not hard to see that not all continuous linear operators on A take the form f(t) in F. 5. SHEFFER SEQUENCE By a sequence Pn(x) in A we shall mean that n ranges over all integers and deg pn(x) = n. Theorem 5,1: Let f(t) be a delta series and let deg g(t) = 0. Then the identity {g(t)f(t)^|s,(x)) = c„ 5„,,, (5.1) Valid for all integers n and k, determines a unique sequence of Laurent series to A. Theorem 5.2: (the Expansion Theorem). Let Sn(x) be Sheffer for (g(t), f(t)). Then for any h(t) in A. k=m ^V. where m = deg h(t). Corollary 1. Let Sn(x) be Sheffer for (g(t), f(t)). Then for any Laurent series p(x) in A. . g(.)f(t)' P(x) p(x)= X ^ ' Sj(x), Where n = deg p(x). Theorem 5.3: The sequence Sn(x) is Sheffer for (g(t), f(t)) if and only if the sequence g(t) Sn(x) is the associated sequence for f(t). Theorem 5,3 says that each associated sequence generates a class of SheiTcr sequences, one for each g(t) of degree 0. We would like to characterize Sheffer sequence in terms of linear operators in r. Theorem 5.4: A sequence pn(x) is the associated sequence for f(t) if and only if (i) Now we may easily derive a characterization of those continuous linear operators on A of the form g(t) in F. Theorem 5.5: A continuous linear operator U on A is of the form g(t) in r if and only if there exists a delta operator f(t) for which Uf(t)p(x) = f(t)Up(x) for all p(x) in A. Theorem 5.6: A sequence Sn(x) is Sheffer for (g(t), f(t)) for some g(t) of degree 0 if and only if f(t)s„(x) = ^s„.,(x) (5.2) for all integers n. Theorem 5.7: A sequence Sn(x) is Sheffer for (g(t), f(t)) for some g(t) of degree 0 if and only if f(t)S.(x)=^s,,(x) for all integers n. Theorem 5.8: Let s(x) be a series in A with deg s(x) = m. Let f(t) be a delta series. Then there is a unique Sheffer sequence Sn(x) with delta series f(t) satisfying Sm(x) = s(x). In fact we have s„(x)=^f(tr s(x) c„ We can characterize Sheffer sequences by an identity, which generalizes the binomial identity. Theorem 5.9: (the Sheffer Identity). A sequence Sn(x) is Sheffer for the pair (g(t), f(t)) for some g(t) if and only if c„ k=-co C|^ C„_j, where pn(x) is the associated sequence for f(t). 13 CHAPTER - V A q-UMBRAL CALCULUS CHAPTER- V A q-UMBRAL CALCULUS L INTRODUCTION A sequence of polynomials {pn(x)} is of binomial type if pn(x) is of exact degree n for all non-negative integers n and {n(x)} satisfies the binomial type theorem ^n^ Pn(>^ + y)=Z P.(x)p„-,(y). (1.1) k=0 vky A sequence of polynomials {pn(x)} is a polynomial set if po=l, and pn(x) is a polynomial of precise degree n, n = 0, 1, 2, ... The combinatorial theory of polynomials of binomial type was developed by Mullin and Rota [152], and in later papers by several authors. We refer the interested reader to the extensive bibliographies in Rota et al. [193] and in Roman and Rota [189]. The analytic theory of these polynomials is much older, see Sheffer [198]. Guinand's work [128] contains an interesting review of the classical Umbral method. Andrews developed this theory in [18]. He introduced the concept of an Eulerian family of polynomials. Definition 1.1: A polynomial set {pn(x)} is an Eulerian family if its member satisfy the functional relationship II Pn(> 14 The Gaussian binomial coefficient is given by -j = 0 if k >n (1.3) (q;q)n if k (a; q)o = 1 and (a; q)n = (1 - a) (1 - aq)... (1 - aq"-')n-K , n > 0. (1.4) n The Gaussian binomial coefficient counts the number of k- k L Jq dimensional subspace of an n-dimensional vector space over a field with q- elements. GF(q). An Eulerian family is {0n(x)}, where eo(x)=l,e„(x)=(x-l)(x-q)...(x-q"-'),n>0. (1.5) Another Eulerian family is '(q;q)n (x -1)". We shall use the notation V n! (a;ql=n (l-aq") (1.6) n=0 A related important set is {6n (x, y)}, with e„(x,y)=l,en(x,y) = (x-y)(x-qy)...(x-q"-'y),n>0n- I . (1.7) Roman and Rota [189] introduced an umbral calculus by introducing a product on P , the dual of the algebra of polynomials over a field of characteristic zero. They defined the product of two functionals L and M by 15 (LM|X") = X (L|X^)(M|X"-J), (1.8) ° [ij and showed that under the usual addition and above multiplication P is a topological algebra. The topology is defined in the following manner. Deflnition 1.2: A sequence {Ln} of linear functional converges to L e P' if and only if for ever p e P, there is an no such that Roman and Rota refer to the algebra P as the umbral algebra. We shall invariably use P and K[x] to denote the same algebra, namely, the polynomials over a commutative integral domain K. One of the purposes of the present work is to construct an umbral calculus that plays in the theory of enumeration of vector spaces over GF(q) the role played by the roman-Rota umbral calculus in the theory of binomial enumeration. In this setting we replace the product (1.8) on P by Section 2 contains the algebraic setting for the polynomials of binomial type. Although the results of Section 2 are not new, the presentation is certainly new. Section 3 contains the umbral lemma and its applications. This is one of the main results of the present paper. The umbral lemma tells us when two polynomials sets are "similar" in the sense that properties of one of them can be deduced from the other. Usually we have a model polynomial set and we would like to identify all the other "similar" polynomial sets. In the roman-Rola 116 umbral calculus the model set is {x"} and the class of similar polynomials is the class of polynomials of binomial type. Section 4 illustrates the intricate interplay of comultiplications in bialgebras, functional relationships like (1.1) and (1.2) and the product (of functionals) on the umbral algebra P*. This applied to derive some properties of the Eulerian families of polynomials. Section 5 contains further results on these polynomials. We attempted to include in the present work only the basic part of a q- umbral calculus with very few applications. Several related important topics, for example, the Lagrange inversion, are still under investigation. In the remaining part of the introduction we include some standard definitions. K shall always denote a commutative integral domain. Definition 1.3: (Tensor Products). Let Vi and V2 be two modules. A tensor product is a pair {^, G) such that (i) (}) is a bilinear mapping f of V] x V2 into the module G. (ii) The range of (j) spans G. (ill) For every bilinear map f of Vi x V2 into a module H there is a map g that maps G into H such that f = gcj). Usually (|)(x, y) is denoted by x ® V2. If V] and V2 are algebras then V| ® V2 equipped with the product (iv) (xi ® yi) (x2 ® y2) = X] X2 ® yi y2, is also an algebra. 117 Definition 1.4: (Tensor Product of Functionals). Let Vi, V2 be two modules over a K. The tensor product Lj ® L2 of the linear functionals Lj and L2 maps Vi ® V2 into K via Similarly, the tensor product of mappings |i and v is (|a ® v) (V| ® V2) = ia(V|)(8)v(V2). Throughout the present paper we shall use "functional" to mean "Linear functional". We now introduce the concept of a bialgebra. Definition L5: (Bialgebras). A bialgebra V is an algebra over K equipped with a multiplication (i:V ® V->V, a unit e:K -^ V, a comultiplication A:V -> V ® V and a counit E : V ^ K satisfying (!) |x (|i ® I) = |a (I ® |i) (associativity), (ii) n (I ® e) ji = [J, (e ® I) J2 = I (e is a unit), (iii) (A ® I) A = (I ® A) A (coassociativity), (iv) J2' (E ® l) A j~' (I ® E) A = I (E is a counit), where I is the identity mapping of V into itself, ji,J2, j",' , J2' are defined by j,(v) = v®l, J2(v)=:l®v, j-'(v®l) = v, j-'(l®v) = v for all V e V and 1 is the multiplicative unit in K. Finally we define symmetric maps. Let V be a vector space over a field K and let J be the map: J(V, ® V2) = V2® V|, Vi,V2eV, (1-12) defined on V ® V. Definition 1.6: (symmetric Maps). A map I of V into V ® V is symmetric if and only if Jol-I. (1.13) Where J is the map defined in (1.12). Definition 1.7: (Ea and rf). The functional 8a and the operator Tf are defined on K [x] by (sjp(x)) = p(a) (1.14) and (r]^p)(x)=p(ax), (1.15) respectively. In two variables the defining formulas are kb|p(^,y)) = p(a,b) (1.16) and (ri^''p)(x,y)=p(ax,by). (1.17) 2. AN ALGEBRAIC SETTING FOR POLYNOMIALS OF BINOMIAL TYPE It has been generally felt that certain families of polynomials {pn (x)} may be formally manipulated as if pn (x) was actually x". thus results that are easy to show for pn(x) = x" would follow in an identical formal way for the other Pn(x)'s yielding less obvioys results. Roman and Rota [189] made this 119 process precise for polynomials of binomial type (pn(x)} by formulating what they termed "the umbral calculus". This formulation works because polynomials of binomial type are modeled after the monomials {x"} and the monomials satisfy the binomial theorem. This formulation then can be carried out in the same way for other polynomials. So, one may show results for these model polynomials and get as a corollary results for any "similar" sets of polynomials. The important contribute of this procedure is not providing results that are necessarily new or deep, but rather unifying seemingly different results in a simple way. In Section 3 we will illustrate what is involved in this procedure by testing the case of Eulerian families of polynomials in which the model polynomials are the 0n's of (1.5). We hope that this will illustrate some aspects of the procedure which go unnoticed in the umbral calculus of Roman and Rota [189] because of the specific form of the model polynomials in that case. As we already mentioned in the introduction, we give a general outline oi' the method by first reviewing the ideas of the umbral calculus in the present section. The model polynomials will have to satisfy functional relationships like (1.1) and (1.2). Our approach can be extended to treat model polynomials satisfying the relationships. PnWx,y)) = 2; aj(y)p^(x)p„_j(y) (2.1) j=o for some polynomial a (x, y). We want to consider pn(x) to be x", n = 0, 1, ..., in some precise manner. This is not too difficult to do because {pn(x)} forms a basis for the algebra 120 K[x], the algebra of polynomials over K. K[x] is endowed with the usual addition and multiplication. We now introduce a new product "*" on the set k[x]. The star product is defined relative to a polynomial set {pn(x)}. Definition 2.1: *^[^] will denote the algebra of polynomials equipped with the usual addition, the usual multiplication by scalars and the star product "*" defined on the given set of polynomials {pn(x)} by Pn*Pm = Pm + n, (2-2) and is extended to •K[x] by linearity, since {pn(x)} is a basis. It is clear that the star product is well defined. We shall adopt the convention of adding a * to a formula involving products when the star product is used instead of the ordinary product. Thus p"*:=p*p*...*p. (2.3) n times Clearly p.=pr (2-4) Observe that the functional equation (1.1) satisfied by polynomials of binomial type {pn (x)} implies Po (x) = 1 and pn (0) = 5n,o- So, at this stage we restrict ourselves to polynomials satisfying Po(x)=l, p,(0) = 0. (2.5) Combining this assumption with (2.3) we arrive at the crucial relationship Pn(>c) = (p,(l)rx-. (2.6) [21 So, pn (x) is (cx)"*, in a suitable multiplication. We must now relate this star multiplication to the usual multiplication in K[x] since a concrete theorem involving the star multiplication will not be very useful without a translation to the language of K[x]. Of course it is obvious that the star multiplication coincides with the ordinary multiplication if and only if pn (x) = (en)", The trivial case. Luckily for us K[x] is equipped with another very natural structure, the comultiplication A mapping K[x] into K[x] ® K[x] in the following manner: A (x) = X ® I + 1 ® X (2.7) and A (p (x)) = p (A (x)), for all p e K [x]. (2.8) Recall that K[x] ® K[x] is an algebra with (qi 0 q2) (r, ® r2) = qi r, ® q2 r2, qj qj, r, rj € K[x]. (2.9) Condition (2.8) just says that A is an algebra map. So, A is completely specified once we require it to be an algebra map and define it symmetrically on a generator. Both properties, i.e., being an algebra map and symmetry, are essential in what follows. Observe that K[x] also happens to be a graded algebra and the comultiplication (2.7), (2.8) is actually grade preserving. This is just a coincidence and is not necessary. In fact the comultiplication A' : x -> X ® x used in the case of Eulerian families of polynomials in Section 3 is not [22 The principal use of the comultiplication is that it can be used to define a product of linear functional on K[x}. Definition 2.2: Let L and M be two linear functional on K[x]. The product functional LM is defined as (LM|p(x)> = {L<8»M|A(p(x))>, for all p e K[x]. (2.10) The above product of linear functionals is commutative since A is symmetric. Definition 2.3: The * product of two linear functionals, say, L and M is defined by (L + M|p(x)) = (L ® M|A' P(X)) , for all polynomials p (x) e *K[x], (2.11) where A* is the comultiplication on »K[x]. A'X = X®1 + 1®X. (2.12) and A is an algebra map. We now observe that although the linear functionals on K[x] do not depend on the product of members of K[x], the products of such functionals, namely (2.10) and (2.11), seem to depend on the product used. So we next ask ourselves the question. When will the products (2.10) and (2.11) be equal? It is clear that this will happen if and only A = A*. When this holds we find ourselves in a very good position because any results that can be formulated using only the products of functionals (2.10) will hold for the star multiplication if they hold for the usual multiplication. This will accomplish the desired result. 123 Preposition 2.4; Let {pn(x)} be a polynomial set satisfying (2.5) and defining a star product. The comultiplications A and A are equal if and only if the sequence of polynomials {pn(x)} is of binomial type. Corollary 2.5: Any sequence of polynomials (pn(x)} that satisfies (2.5) also satisfies ^nA A'(P,W)=I Pj(x)®P„-j(x). (2.13) j=0 vjy We combine (2.13) and the definition of product of functional to establish. Corollary 2.6: Where a sequence of polynomials {pn(x)} that satisfies (2.5), the product of functionals L and M is given by /.,^ LM|p„(x)) = X L|PJ(X)){M|P„_^(X)), (2.14) j=0 u; then (2.14) holds whenever {pn(x)} is a sequence of polynomials of binomial type. Corollary 2.7: (Roman and Rota [189]). If the product of functionals on K[x] is defined by ^n^ LM x>)iM .n-J (2.15) j=0 U; then (2.14) holds whenever {pn(x)} is a sequence of polynomials of binomial type. 124 Theorem 2.8: (Expansion Theorem). Let {pn(x)} be a polynomials set of binomial type. Then PW = S -!i(Lip(x))pj(x), (2.16) 0 J where L|PnW} = S„l 3. THE UMBRAL LEMMA AND APPLICATIONS We now generalize the results of Section 2 to almost any family of polynomials. The umbral lemma that almost any sequence of polynomials may be considered the same as any other sequence as long as we are allowed to alter the multiplication. This will allow us to alter formulas involving one sequence of polynomials to find formulas for other sequences. Definition: The polynomial p*(x) associated with a polynomial p(x) and a multiplication * is defined by p*(x)=XcjX^* iff p(x)=XCjX\ (3.1) Theorem 3.1: (The Umbral Lemma). Let {pn (x)} be two sequences of polynomials such that the nth polynomials is of precise degree n. then (a) and 9b) are equivalent. (a) There is a unique multiplication * on K[x] such that (i) Pn(x)=b|;(x),n = 0,l,.... 125 (ii) There exists a mapping S : (K[x],.) -> (K[x], *), called the starring map, such that S is an isomorphism and S(x) = x. S(I) = 1. (b) po(x) = bo(x) and p,(x) - b,(x). The map S of the umbral lemma will be called the starring map and its inverse will be called the star erasing map. Next we investigate consequences of the umbral lemma. Our first result is Theorem 3.2: Let {pn(x)} be any polynomial set. Then exists a product * such that Pn(x) = (c(x-a))"\ (3.2) where pi(x) = c(x - a). Note that for any polynomial set {pn(x)} there is no loss of generality in assuming po(x) = 1. When a = 0 in Theorem 3.2 we are lead to polynomials more general than the polynomials of binomial type. What forces these polynomials to be a binomial type is the comultiplication structures associated with A and A , as we saw in Section 2. Our next result covers the Eulerian families of polynomials, but unfortunately we have to go to two variables in order to handle them. Theorem 3.3: Let {pn(x, y)} be a sequence of polynomials such that pn (x,y) is homogeneous of degree n and Po(x, y)=l, p, (x, y) = c(x-y). (3.3) then there is a product * on R [x, y] so thai Pn(x,y) = c"(x-y)*(x-qy)*...*(x;^'y), n>l. (3.4) 126 Definitions 3.4: By an operator U we mean a mapping of K[x] to K[x]. An operator U is a degree reducing operator if the degree of Ux" is n - 1. Tlie functionals eg and e* are defined via (Ba|p(x)> = Zc^a\ (s;|p(x)) = Xs^' (3-5) j j when p(x)= Z Cj x^ eK[x]. The degree reducing operator expansion theorem is Theorem 3.5: (Degree Reducing Operator Expansion). Let {pn(x)} be a polynomial set. Then there is a degree reducing operator U and a functional L such that the following expansion holds: PW = Z (L|u"p(x))p{x)/n!, p(x)eK[x]. (3.6) 0 Furthermore if (e^ |Pn(x)) vanishes for all positive n then L = Eg. For polynomials of binomial type {pn(x)} the functional L is SQ since Pn(0) = 5n_ 0- When {Pn(x)} is an Eulerian family of polynomials pn (1) is &„ o, hence L coincides with E]. Theorem 3.5 implies another expansion theorem, namely, Corollary 3.6: For every polynomial set {pn(x)} with po(x) = 1 there exists a functional L and a degree reducing operator U such that any functional or operator A has the expansion A. = g :;^Piifc)(L|u"-). (3.7) 127 In Theorem 3.5 we discovered that a polynomial set {pn(x)} determines a degree reducing operator U and an expansion (3.7). We now show that the converse is also true, that is, U and (3.7) essentially determine the pn's. Theorem 3.7: Given a degree reducing operator U and a sequence of scalars {an} there exists a unique polynomial set {pn(x)} and a unique functional L such that p„(a„) = 0, n>0, (3.8) and (3.7) hold. Definition 3.8: A polynomial set {pn(x)} is associated with a degree reducing operator U if. Upn(x) = up„_,(x). (3.9) Let us reexamine the role played by the sequence {an} in Theorem 3.7 and its proof. The sequence was used to exhibit a functional M defined on the Pn's by {M|Pn(x)) = 5,,o. (3.10) and extended to all of K[x] by linearity. This functional enables us to compute the constant term in pn(x) once all the other terms have been already computed from (3.9). The converse is also true in the sense that every polynomial set uniquely determines a degree reducing U satisfying (3.9) and generates a functional M via (3.10). This proves 128 Theorem 3.9: There is a one-to-one correspondence between polynomial sets {pn(x)} and pairs {U, M}, U is a degree reducing operator and M is a functional. Furthermore (3.9) and (3.10) hold. When the polynomials pi(x), p2(x), ... have a common zero a, M is Sg. for polynomials of binomial type a == 0 while a = 1 for Eulerian families of polynomials. Our next result is an expansion theorem in polynomials that resemble the 6n's. Theorem 3.10: (Operator Expansion Theorem). Let {pn(x)} be a polynomial set with pi (x) = x - 1. There exists an operator U and a functional L so that p(x)=X %^(-ir(L|e„(U)p(x))p„(x). (3.11) 0 lq;qjn Further if pn (1) = 6n,othen L = ej. Note that the operator expansion (3.11) is easy to prove because all it uses is (3.12). What is nontrivial is finding a functional expansion where the coefficients of pn(x) are scalar multiplies of (r^(L)|p(x)) for some seque4nce of polynomials {rn (L)} and a certain functional L. In the next section we shall see that it is impossible to do so for Eulerian families of one variable. This situation can be remedied if we go to homogeneous polynomials in two variables. 129 4. EULERIAN FAMILIES OF POLYNOMIALS In the present section we explore certain properties of Eulerian families of polynomials via the comultiplication A:x->x®x. (4.1) As we pointed out in Section 2 the comultiplication induces a product on P*, the dual of K[x]. The product of functionals is {LM|P(X)) = (L®M|P(AX)). hence LM x") = (L®M (X®X)") = (L®M x"®x' Consequently LM X" = L X")(M (4.2) We now characterize Eulerian families of polynomials in terms of functional products. Recall that a polynomial set {pn(x)} is an Eulerian family if and only if Pn(xy)=Z Pk(x)y'Pn-k(y)- (4.3) Theorem 4.1: A polynomial set {pn(x)} is an Eulerian family if and only if n n Pn(x)) = I {L|P,(X))(MX^P„_,(X)' (4.4) 0 L k jq 130 Theorem 4.1 illustrates the connection between the functional relationships (4.3) and the product of functional (4.2). For a given Eulerian family {pn(x)} set Pn(x)=Zc„,,x™, n=0,I,.. (4.5) Theorem 4.2: We have n ^n,m ~ Cm,mPn-m(0)- (4.8) m Corollary 4.3: m dm.n = c / c (4.11) m-n.m-n m,m n Observe that the relationship (4.11) determines d^n uniquely when the sequence {Cn,n} of leading terms in the p's is given. The uniqueness of the polynomials then follows from the uniqueness of the d's and simple induction since (recall (4.6) and (4.11)) m m c x"" = E ^m-n,m-n PnV^J- (4.12) n=0 n Relationship (4.12) solves the connection coefficient problem expressing the monomials in terms of members of an Eulerian family of polynomials. The n connection coefficients ^m-n m-n ^avc a vcry simple form indeed. The m -Jq inverse problem, namely, expressing the p's in terms of the monomials, seems 13- to be much harder. Luckily generating functions come to the rescue because a generating function is really an infinite family of identities. The next result characterizes Eulerian families of polynomials in terms of their generating functions. Theorem 4.4: (Andrews [18]) A polynomial sequence {pn(x)} with Po(x) = 1 is an Eulerian family if and only if it has a generating function where f(t) = f; yn^-^"- Yo=l> Yn^O, n = l,2,..., (4.14) 0 (q;q)n and the leading term Cn,n in Pn(x) is Yn. Andrews [18] proved Theorem 4.4 in a different way. As a matter of fact, our approach is completely different. We now investigate the Eulerian family {6n(x)}. Let X, Y, Z be vector subspaces of V^, the n-dimensional vector space over GF(q). Assume that X, Y, Z contain x, y, z vectors, respectively. Goldman and Rota showed that 6n (x, y) counts the number of one-of-one linear transformations f of Vn into X such that f(x) n Y == {0}, and Y is a subspace of . By a eleven counting argument that squeezes a third subspaces Z between the subspaces X, Y Goldman and Rota proved that en(x,y) = S e,(x,z)G„_,(z,y).. (4.15) From this one can easily use the obvious identity en(x, y) - y" n(x/y) 132 to show that the polynomials 6n(x) satisfy (4.3), hence {9n(x)} is an Eulerian family. Andrews gave a combinational argument, different from Goldman and Rota, to show that {9n(x)} is an Eulerian family. The polynomials {6n(x)} are monic, thus (4.12) reduces to m m "^ = Z e.(x). (4.16) n=0 L n -iq We now determine the inverse relation to (4.16), namely, the following. Theorem 4.5: (Gauss' Binomial Theorem). The polynomials 0n(x) are given explicitly by n en(x)=I -1) n-m q^(n-m)(n-m-l)/ 2 m (4.16) m=0 m Theorem 4.6: (Heine's Binomial theorem). We have z (^'^X „n _ (ax;q)„ (4.18) (q;q)n (>';qX Euler's formula (4.19) can be proved analytically by showing that both sides satisfy the functional equation. f(x)-f(qx)^^^^^^ ^^^^.^^ f(x) = lfe) X 1-X and are continuous at x = 0. A truly combinational proof using Ferrer's diagram for partitions of integers is in Andrews. The sum (4.18) is an important result in special functions. i33 Theorem 4.7: (Expansion in terms of 0n(x)). The expansion of an arbitrary polynomial p(x) in terms of the 0's is P(x)=Z ^(^(-0"(e„(e<,|p(x))e„(x), (4.21) and 9o(eq) is interpreted as 8]. Since our model polynomials are {6n(x)}, the expansion formula (4.21) suggests replacing {6n(x)} by any Eulerian family of polynomials {pn(x)} and replacing 8q by some other functional L so 6n(Sq) becomes On(L). Unfortunately this is not the case, and (4.21) or (4.22) in effect characterize the 6n(x)s up to scaling factors. Theorem 4.8: The only functional L and Eulerian family {pn(x)} thai satisfy {0n(L)|p.(x)> = a„5„,„ (4.24) for some sequence of nonzero constants {an} are L = 8q and Pn(x) '-^ k" 6n(x)- where k = ai/(q - 1). Furthermore an is given by a„=(q;q).q"'"-""af/(l-q)". In the next section we will extend the functional expansion (4.21) to expansion in terms of Eulerian families of polynomials by constructing a two variable umbral calculus. 134 5. HOMOGENEOUS EULERIAN FAMILIES We start this section by developing a two variable umbral calculus that will yields the functional expansion for Eulerian families of polynomials. Definition 5.1: Define the comultiplication A on the generators x and y of R[x, y] by Ax = qx® 1, Ay= 1 ® y (5.1) and extend it to R[x, y] using Ap (x, y) = p(Ax, Ay). (5.2) The above definition induces the functional product (LM|p(x,y)) = L®M| p(Ax,Ay) ). (5.3) Our model polynomials are the two variable polynomials 6n(x, y) introduced in (1.7). Let us compute the action of a product of two functionals on 0n(x, y). Clearly LM|e,(x,y)) = (L®M|e„(Ax,Ay)) n n = (L®MIy e,(qx®ll®l)e„_.(l®l,l®y) 0 u-'J.J. q n n = L®MXI . e^(qx,l)®e„_^(l,y) 0 J.J q Therefore LM|e„(x,y)) = 2: {L|e,(qx,l))(M|e,_^(l,y)}. (5.4) 135 The above product of functionals is not commutative. Set en(L):=Z (_i)''-iq(<'-.i)(«-H)/2L^ (5.5) L'-s,,,, L' = L, V = V-'L forj>l. (5.6) Theorem 5.2: Under the above functional multiplication we have e„k,)|e„(x,y)) = (-ir(q;q)„q"<"-')'^5„,„ (5.7) Definition 5.3: (Andrews [18]). We say that {pn(x, y)} is a homogenous Eulerian family of polynomials if each Pn(x, y) is a homogeneous polynomial of degree n in x and y po(x, y) = I, pn(x, 0) ;^ 0 and II Pn(X'y)=Z Pn(X'Z)Pn-k(z>y) (5.9) L Jq Observe that homogeneous Eulerian families of polynomials satisfy Pn(x,x) = 0 (5.10) as can be seen from (5.9). The polynomials Gn(x, y) form a homogeneous Eulerian families of polynomials. Lemma 5.4: Let {pn(x, y)} be a homogeneous Eulerian family of polynomials with pi(x, y) = x - y. then there exists an isomorphism S from (K[x],.) to (K[x], *), K: = R[y] such that e:(x,y)-S(e„(x,y)) = p„(x,y). (5.11) Furthermore we have 136 6n(a,y)=en(a,y), e'„(x,a)=p„(x,a), 0*„(ax,y)=p„(ax,y), for all a € K = R[y]. (5.12) We now extend the functional expansion theorem (theorem 4.7) to homogeneous Eulerian families of polynomials. Theorem 5.5: If {pn(x, y)} is a homogeneous Eulerian family of polynomials then there exists an associated functional M such that (0n(M)|pJx,y)) = (-l)"(q;q)„q"^"-'^"6^,, (5.13) where the functional multiplication is as in (5.3) and (5.6). Theorem 5.6: Define A(x) = qx ® 1 and Ap(x) = p(A(x)). If Pn(x) is an Eulerian family of polynomials there is a functional L such that '^' ^ H)"(q;q)„q"'"""'"*^- 137 BIBLIOGRAPHY [1] N. ABDUL HALIM and W. A. AL-SALAM, A characterization of Laguerre Polynomials, Rend. Sem. Mat. Univ. Padova XXXIV (1964), 176-179. [2] W. R. 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