Discriminants: calculation, properties, and connection to the root distribution of polynomials with rational generating functions

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 2 New discriminant calculations: Triangular numbers and a di-
agonal sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Triangular numbers and Chebyshev polynomials . . . . . . . . . . . . . . . . 4
2.2 A linear polynomial transformation and its root distribution . . . . . . . . . 7
2.3 The diagonal sequence of a resultant . . . . . . . . . . . . . . . . . . . . . . 9
Chapter 3 A property of q-discriminants of certain cubics . . . . . . . . . 18
3.1 Factorization of the q-Discriminant . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Sensitive Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 4 Factorization of discriminants of transformed Chebyshev poly-
nomials: The Mutt and Jeff syndrome . . . . . . . . . . . . . . . . . . . 32
4.1 Discriminant, resultant and Chebyshev polynomials . . . . . . . . . . . . . . 34
4.2 The Mutt and Jeff polynomial pair . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 The discriminant of J(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.4 The discriminant of M(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 The roots of M(x) and J(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Chapter 5 Roots of polynomials and their generating functions: A specific
example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 A general form of the discriminant . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Generating function for H (1)
m (q)
. . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 Generating function for H (2)
m (x) . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.4 A hypergeometric identity from Euler’s contiguous relation and the Wilf-
Zeilberger algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.5 Generating function for H (n)
m (x) . . . . . . . . . . . . . . . . . . . . . . . . .
63
Chapter 6 Roots of polynomials and their generating functions: A general
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93