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 i 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 . 27 2.2.1 Translation Operators . 28 2.2.2 Basic Sequences and Delta Operators . 30 2.2.3 Special Cases . 32 2.2.4 Exercises . 40 2.3 Sheffer Sequences . 44 2.3.1 Initial Values Along a Line . 47 2.3.2 The Umbral Group . 51 2.3.3 Special Cases . 54 2.3.4 Exercises . 63 2.4 Transfer Theorems . 69 2.4.1 Umbral shifts and the Pincherle derivative . 73 2.4.2 Proof of the Transfer Formula . 74 2.4.3 Exercises . 76 ii CONTENTS 3 Applications 79 3.1 The Functional Expansion Theorem . 79 3.1.1 Some Applications of the Functional Expansion Theorem . 83 3.1.2 Exercises . 87 3.2 Diagonals of Riordan Matrices as Values of Sheffer Sequences . 89 3.2.1 Exercises . 93 3.3 Determinants of Hankel Matrices . 94 3.3.1 Exercises . 100 3.4 Classical Umbral Calculus . 102 3.4.1 The Cumulant Umbra . 109 3.4.2 Exercises . 111 4 Finite Operator Calculus in Several Variables 113 4.1 Polynomials and Operators in Several Variables . 114 4.1.1 Exercises . 117 4.2 The Multivariate Transfer Formulas . 119 4.2.1 Transfer with constant coeffi cients . 119 4.2.2 Operator based transfer . 121 4.2.3 The multivariate Pincherle derivative . 123 4.2.4 Transfer with operator coeffi cients . 128 4.2.5 Exercises . 131 4.3 The Multivariate Functional Expansion Theorem . 134 4.3.1 Exercises . 136 5 Special Constructions in Several Variables 137 5.1 Multi-indexed Sheffer Sequences . 137 5.1.1 Delta Operators for multi-indexed Sheffer sequences. 139 5.1.2 Translation invariance of diagonalization, and some examples.140 5.1.3 Abelization of Multi-Indexed Sequences . 142 5.1.4 Exercises . 147 5.2 Polynomials with all but one variable equal to 0 . 150 5.2.1 Exercises . 153 5.3 Cross-Sequences and Steffensen Sequences . 153 5.3.1 Exercises . 155 6 A General Finite Operator Calculus 157 6.1 Transforms of Operators . 157 6.2 Reference Frames, Sheffer Sequences, and Delta Operators . 162 6.2.1 Reference Frames . 162 6.2.2 Sheffer Sequences and Delta Operators . 165 6.2.3 Exercises . 169 6.3 Transfer Formulas . 171 6.3.1 General Umbral Shifts and the Pincherle Derivative . 171 6.3.2 Equivalent Transfer Formulas . 173 CONTENTS iii 6.3.3 Exercises . 175 6.4 Functionals . 176 6.4.1 Augmentation . 179 6.4.2 Orthogonality . 179 6.4.3 Exercises . 184 7 Applications of the General Theory 187 7.1 The Binomial Reference Frame . 188 7.1.1 Orthogonal Binomial Reference Sequences . 189 7.1.2 Generalized Catalan Operators . 195 7.1.3 Dickson Polynomials . 197 7.1.4 Exercises . 200 7.2 Eulerian Differential Operators . 204 7.2.1 Exercises . 209 8 Solutions to Exercises 211 Bibliography 257 iv CONTENTS 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 n Every linear operator T on polynomials has a representation T = n 0 M(pn) , ≥ D where is the derivative operator, and M(pn) stands for multiplication by the D P polynomial pn (x) (Pincherle [74, 1901]). When applied to polynomials, the oper- ator T reduces to a finite sum, of course, and may therefore be called a “finite” n operator. A special case of this concept, when T can be written as T = n 0 cn , ≥ D where c0, c1,... are scalars, is called a called a finite operator in the “Finite Op- erator Calculus” [83, 1973] by G.-C. Rota and his students D. KahanerP and A. n Odlyzko. Hence T is isomorphic to the formal power series n 0 cnt , and it is of course exactly this isomorphism that made the Finite Operator≥ Calculus so widely applicable. We adopt Rota’s approach in this book, but consider,P in the last two n chapters, also linear operators of the form T = n 0 cn , where can be any linear operator reducing the degree by 1, deg ( q) =≥ deg qR 1 for all polynomialsR q of degree larger than 0, and q = 0 for polynomialsR P of degree 0 (we identify polyno- mials of degree 0 with scalarsR c = 0; however, we let deg (0) = , as usual). The fundamental difference between6 Rota’s approach and the generalized1 version we n present, due to J. M. Freeman [35, 1985], also his student, is that T = n 0 cn is translation invariant, TEc = EcT , where Ec : f (x) f (x + c) is the operator≥ D translating by c. The generalized version does not have7! this property.P Translation invariance, however, is an important feature in many applications. The applications we have in mind are usually the solutions to recursive equa- tions. Suppose your analysis of a given enumerative problem resulted in a recursive expression for the numbers you are looking for; to be specific, let us assume you arrived at Fm = Fm 1 + Fm 2, the recursion for the Fibonacci numbers, starting at F0 = F1 = 1. A computer will give us “special”answers in a very short time, even F10,000 makes no problem whatsoever. From this point of view, you will not need this book. However, finding n+1 n+1 n+1 out that Fn = 1 + p5 1 p5 / 2 p5 is as surprising as it is rewarding. In addition, it is even “practical”; a scientific calculator will show that 2089 F10,000 5.4 10 . It also tells you something about the ratio, Fn/Fn 1, and CONTENTS v 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 first 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 Fn (m) = Fn (m 1) + Fn 1 (m 2) , with initial conditions Fn (n) = Fn 1 (n) for n 1, and F0 (m) = 1 for all m. The recursion is still easy, but the initial values≥ are also “recursive” - we have to know Fn 1 (n) before we can say what the value of Fn (n) is. We will see that F0 (x) ,F1(x),... is a sequence of polynomials, actually a basis, and that the operator T : Fn (x) Fn 1 (x) is a translation invariant operator, satisfying the operator equation 7! 1 2 I = E + E T. We will show in chapters 2 and 3.1 how to find the solution with given initial values from such an equation. Suppose you found the system of recursion sm,n (u, v) = sm,n (u 1, v) + sm 1,n (u, v + 2) and sm,n (u, v) = sm,n (u, v 1) + sm,n 1 (u + 1, v + 1) sm,n 2 (u, v) , with initial values sm,n (0, 0) = 0 for all m, n 0, except s0,0 (0, 0) = 1. This system of recursions is two dimensional and linear,≥ the initial values are explicit. We show how to write sn,m (u, v) as a sum of binomial coeffi cients in chapters 3.1 and 5, dedicated to the multivariate Finite Operator Calculus. Technically the problems get more diffi cult to solve, of course, when it comes to higher dimensions. The theory, however, remains quite easy. Somewhere between the univariate and the multivariate case fall the Steffensen sequences (section 5.3) and the multi- indexed sequences (section 5.1). Rota’s goal was to create a solid foundation for the “Umbral Calculus”, to purge it of the “witchcraft”, as he called it. This was one of his favorite themes, and he wrote more papers on Umbral Calculus later. Several young (at that time) 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 “Finite Operator Calculus”was published a part VIII in a series of papers “On the Foundations of Combinatorial Theory”. A complete survey of papers relating to Umbral Calcu- lus until the year 2000 has been compiled by Di Bucchianico and Loeb [26]. An application of the Finite Operator Calculus can also be found in Taylor [96, 1998]. We assume that after 5 chapters the reader will get interested in the theory itself. J. M. Freeman explored this generalization in some depth [35], and we follow it literally in the last two chapters.