Contents
Foreword vi
Introduction vii
Chapter 1 The Poset Conjecture 1
1.1 Introduction 1
1.2 Statement of the Poset Conjecture 1
1.3 Partial results on the Poset Conjecture 4
1.4 A second look at the Poset Conjecture 6
Chapter 2 A General Theory 8
2.1 Introduction 8
2.2 Polynomials with only real zeros and total positivity 8
2.3 The six fundamental bases and the transition matrices
between them 12
2.4 Linear transformations that preserve the PF property 16
2.5 Linear transformations that preserve the PF
2
property 21
2.6 Summary of results and open problems 24
Chapter 3 Ramifications of the General Theory 27
3.1 Introduction 27
3.2 Further properties of the six fundamental bases 27
3.3 The special case of polynomials with no constant term 30
3.4 Eulerian, Lagrange, and Krawtchouk polynomials 31
3.5 Three general problems 35
Chapter 4 Polynomials in P F
fx+d-i ^
39
IA
d
4.1 Introduction 39
4.2 Elementary properties of PF [ ( ^ ~ 1 ) ] 39
4.3 A Fundamental Theorem 41
4.4 Consequences of the Fundamental Theorem 43
4.5 The characterization of PF-sequences 44
4.6 The characterization of PF[(*+*~*)] 48
4.7 Products of polynomials in P F [ ( ^ J " ' ) ] 50
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