Finite Operator Calculus With Applications to Linear Recursions

233 Pages·2013·1.317 MB·English

by  Heinrich Niederhausen

Finite Operator Calculus
With Applications to Linear Recursions
Heinrich Niederhausen
Florida Atlantic University
Boca Raton
[email protected]
www.math.fau.edu/Niederhausen
2
Contents
Contents 3
1 Prerequisites from the Theory of Formal Power Series 1
1.1 Generating Functions and Linear Recursions. . . . . . . . . . . . . 2
1.1.1 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Composition and Inverses . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Multivariate Power Series . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Finite Operator Calculus in One Variable 21
2.1 Polynomials, Operators, and Functionals . . . . . . . . . . . . . . . 21
2.1.1 The Vector Space of Polynomials, and Their Bases . . . . . 22
2.1.2 Standard Bases and Linear Operators . . . . . . . . . . . . 25
2.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Finite Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2.1 Translation Operators
Solving Recursions . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Basic Sequences and Delta Operators . . . . . . . . . . . . 31
2.2.3 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 She⁄er Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.1 Initial Values Along a Line . . . . . . . . . . . . . . . . . . 48
2.3.2 The Umbral Group . . . . . . . . . . . . . . . . . . . . . . . 53
2.3.3 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Transfer Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4.1 Umbral shifts and the Pincherle derivative . . . . . . . . . . 75
2.4.2 Proof of the Transfer Formula . . . . . . . . . . . . . . . . . 76
2.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 CONTENTS
3 Applications 81
3.1 The Functional Expansion Theorem
Functional Initial Values . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1.1 Some Applications of the Functional Expansion Theorem . 85
3.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.2 Diagonals of Riordan Matrices as Values of She⁄er Sequences . . . 92
3.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3 Determinants of Hankel Matrices . . . . . . . . . . . . . . . . . . . 97
3.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.4 Classical Umbral Calculus . . . . . . . . . . . . . . . . . . . . . . . 105
3.4.1 The Cumulant Umbra . . . . . . . . . . . . . . . . . . . . . 112
3.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4 Finite Operator Calculus in Several Variables 117
4.1 Polynomials and Operators in Several Variables . . . . . . . . . . . 118
4.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.2 The Multivariate Transfer Formulas . . . . . . . . . . . . . . . . . 123
4.2.1 Transfer with constant coe¢ cients . . . . . . . . . . . . . . 123
4.2.2 Operator based transfer . . . . . . . . . . . . . . . . . . . . 125
4.2.3 The multivariate Pincherle derivative. . . . . . . . . . . . . 127
4.2.4 Transfer with operator coe¢ cients . . . . . . . . . . . . . . 132
4.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.3 The Multivariate Functional Expansion Theorem . . . . . . . . . . 138
4.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
5 Special Constructions in Several Variables 141
5.1 Multi-indexed She⁄er Sequences . . . . . . . . . . . . . . . . . . . 141
5.1.1 Delta Operators for multi-indexed She⁄er sequences. . . . . 143
5.1.2 Translation invariance of diagonalization, and some examples.144
5.1.3 Abelization of Multi-Indexed Sequences . . . . . . . . . . . 146
5.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.2 Polynomials with all but one variable equal to 0. . . . . . . . . . . 154
5.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.3 Cross-Sequences and Ste⁄ensen Sequences . . . . . . . . . . . . . . 157
5.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6 A General Finite Operator Calculus 161
6.1 Transforms of Operators . . . . . . . . . . . . . . . . . . . . . . . . 161
6.2 Reference Frames, She⁄er Sequences, and Delta Operators . . . . . 166
6.2.1 Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . 166
6.2.2 She⁄er Sequences and Delta Operators . . . . . . . . . . . . 169
6.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.3 Transfer Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.3.1 General Umbral Shifts and the Pincherle Derivative . . . . 175
CONTENTS 5
6.3.2 Equivalent Transfer Formulas . . . . . . . . . . . . . . . . . 177
6.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.4 Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.4.1 Augmentation. . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.4.2 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7 Applications of the General Theory 193
7.1 The Binomial Reference Frame . . . . . . . . . . . . . . . . . . . . 194
7.1.1 Orthogonal Binomial Reference Sequences . . . . . . . . . . 195
7.1.2 Generalized Catalan Operators . . . . . . . . . . . . . . . . 201
7.1.3 Dickson Polynomials . . . . . . . . . . . . . . . . . . . . . . 203
7.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.2 Eulerian Di⁄erential Operators . . . . . . . . . . . . . . . . . . . . 210
7.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
Bibliography 217
7.3 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
6 CONTENTS
CONTENTS i
Preface
The following text originated from various lecture notes for graduate courses in
combinatorics. I would very much appreciate receiving your comments, additions
and corrections.
My apologies to all those mathematicians not mentioned in the text, their
important contributions to the theory of the Finite Operator Calculus skipped
over because of ignorance or by design in order to keep the material manageable.
Your views can still be included - please let me know: [email protected].
Introduction
EverylinearoperatorT onpolynomialshasarepresentationT = M(p ) n,
n 0 n D
where is the derivative operator, and M(p ) stands for multiplic(cid:21)ation by the
n
D P
polynomial p (x) (Pincherle [77, 1901]). When applied to polynomials, the oper-
n
ator T reduces to a (cid:133)nite sum, of course, and may therefore be called a (cid:147)(cid:133)nite(cid:148)
operator.Aspecialcaseofthisconcept,whenT canbewrittenasT = c n,
n 0 nD
where c ;c ;::: are scalars, is called a called a (cid:133)nite operator in the (cid:147)Fi(cid:21)nite Op-
0 1
P
erator Calculus(cid:148)[86, 1973] by G.-C. Rota and his students D. Kahaner and A.
Odlyzko.HenceT isisomorphictotheformalpowerseries c tn,anditisof
n 0 n
courseexactlythisisomorphismthatmadetheFiniteOperator(cid:21)Calculussowidely
P
applicable. We adopt Rota(cid:146)s approach in this book, but consider, in the last two
chapters, also linear operators of the form T = c n, where can be any
n 0 nR R
linearoperatorreducingthedegreeby1,deg( q)=(cid:21)degq 1forallpolynomialsq
RP (cid:0)
ofdegreelargerthan0,and q =0forpolynomialsofdegree0(weidentifypolyno-
R
mialsofdegree0withscalarsc=0;however,weletdeg(0)= ,asusual).The
6 (cid:0)1
fundamental di⁄erence between Rota(cid:146)s approach and the generalized version we
present,duetoJ.M.Freeman[35,1985],alsohisstudent,isthatT = c n
n 0 nD
is translation invariant, TEc =EcT, where Ec :f(x) f(x+c) is the o(cid:21)perator
7! P
translating by c. The generalized version does not have this property. Translation
invariance, however, is an important feature in many applications.
Theapplicationswehaveinmindareusuallythesolutionstorecursiveequa-
tions.Supposeyouranalysisofagivenenumerativeproblemresultedinarecursive
expression for the numbers you are looking for; to be speci(cid:133)c, let us assume you
arrived at
F =F +F ;
m m 1 m 2
(cid:0) (cid:0)
the recursion for the Fibonacci numbers, starting at F = F = 1. A computer
0 1
willgiveus(cid:147)special(cid:148)answersinaveryshorttime,evenF makesnoproblem
10;000
whatsoever.Fromthispointofview,youwillnotneedthisbook.However,(cid:133)nding
out that F = 1+p5 n+1 1 p5 n+1 = 2n+1p5 is as surprising as it is
n
(cid:0) (cid:0)
rewarding. In a(cid:16)dd(cid:0)ition, it(cid:1)is even(cid:0)(cid:147)practi(cid:1)cal(cid:148);(cid:17)a s(cid:0)cienti(cid:133)c c(cid:1)alculator will show that
F 5:4 102089. It also tells you something about the ratio, F =F , and
10;000 n n 1
(cid:25) (cid:2) (cid:0)
ii CONTENTS
its famous limit, the Golden Ratio. Generating functions are the standard tool
for solving this type of linear recursion. We give a brief introduction in the (cid:133)rst
chapter. The reader familiar with formal power series may just want to browse
through section 1.2 for the notation.
Now suppose the recursion you found was
F (m)=F (m 1)+F (m 2);
n n n 1
(cid:0) (cid:0) (cid:0)
with initial conditions F (n) = F (n) for n 1, and F (m) = 1 for all m.
n n 1 0
The recursion is still easy, but the(cid:0)initial values(cid:21)are also (cid:147)recursive(cid:148)- we have
to know F (n) before we can say what the value of F (n) is. We will see
n 1 n
that F (x);(cid:0)F (x);::: is a sequence of polynomials, actually a basis, and that the
0 1
operator T : F (x) F (x) is a translation invariant operator, satisfying the
n n 1
operator equation 7! (cid:0)
I =E 1+E 2T:
(cid:0) (cid:0)
We will show in chapters 2 and 3.1 how to (cid:133)nd the solution with given initial
values from such an equation.
Suppose you found the system of recursions
s (u;v)=s (u 1;v)+s (u;v+2) and
m;n m;n m 1;n
(cid:0) (cid:0)
s (u;v)=s (u;v 1)+s (u+1;v+1) s (u;v);
m;n m;n m;n 1 m;n 2
(cid:0) (cid:0) (cid:0) (cid:0)
with initial values s (0;0) = 0 for all m;n 0, except s (0;0) = 1. This
m;n 0;0
(cid:21)
system of recursions is two dimensional and linear, the initial values are explicit
in this case. We show how to write s (u;v) as a sum of binomial coe¢ cients
n;m
in chapters 3.1 and 5, dedicated to the multivariate Finite Operator Calculus.
Technically the problems get more di¢ cult to solve, of course, when it comes to
higher dimensions. The theory, however, remains quite easy. Somewhere between
theunivariateandthemultivariatecasefalltheSte⁄ensensequences(section5.3)
and the multi-indexed sequences (section 5.1).
Rota(cid:146)s goal was to create a solid foundation for the (cid:147)Umbral Calculus(cid:148), to
purge it of the (cid:147)witchcraft(cid:148), as he called it. This was one of his favorite themes,
andhewrotemorepapersonUmbralCalculuslater.Severalyoung(atthattime)
mathematicians took up his work, and showed that it had applications to a va-
riety of mathematical topics, including approximation theory, signal processing,
probability theory, and, of course, combinatorics. After all, the (cid:147)Finite Operator
Calculus(cid:148)was published a part VIII in a series of papers (cid:147)On the Foundations of
Combinatorial Theory(cid:148). A complete survey of papers relating to Umbral Calcu-
lus until the year 2000 has been compiled by Di Bucchianico and Loeb [26]. An
applicationoftheFiniteOperatorCalculuscanalsobefoundinTaylor[99,1998].
We assume that after 5 chapters the reader will get interested in the theory
itself.J.M.Freemanexploredthisgeneralizationinsomedepth[35],andwefollow
it literally in the last two chapters.
Finally, the expert may wonder why Finite Operator Calculus and not Um-
bral Calculus? Umbral Calculus is a highly (cid:147)symbolic(cid:148)language; no operators,
CONTENTS iii
just umbrae! As a rule of thumb, every Finite Operator statement gets shorter in
Umbral Calculus; for example, the contents of section 2.2.3, Basic sequences with
polynomial coe¢ cients, is reduced to (cid:11)~ (cid:31):(cid:11) in umbral notation. However, it
D
(cid:17)
may be exactly this brevity, achieved through a multitude of special de(cid:133)nitions,
thatpreventsUmbralCalculusfrombeingwidelyknown.Wegiveanintroduction
to Umbral Calculus in section 3.4.
Most of the examples and exercises in this book refer to combinatorial prob-
lems, with few exceptions. Yet this is not a book on enumerative combinatorics,
because we begin where combinatorics ends, that is at the crucial point, where
combinatorics has delivered a recursion and initial conditions. Without question,
this is the hard part of combinatorics; solving the recursion is the technical part.
Finite Operator Calculus can help with the technical part, when applicable. We
will therefore describe the combinatorics only brie(cid:135)y, just enough to introduce
the recursion. There is a special sequence of examples taken from [87] about the
enumeration of lattice paths containing patterns; they are Example 1.2.3, 2.3.15,
2.4.5, 5.1.6, and 5.1.8.
The reader of this text should have access to a computer algebra package
(CAS) like Mathematica, Maple, MuPad, etc. This will allow for checking coe¢ -
cientsofformalpowerseries,gettingconjecturesonnewresults,andverifyingthe
polynomial formulas in the examples and exercises.
iv CONTENTS