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