Some Recent Work on Hankel Determinants

The purpose of this series of lectures is to present some recent work related to Hankel determinants. The initial motivation came from a surprising result due to Allouche, Peyrière, Wen and Wen saying that all Hankel determinants of the Thue-Morse sequence are nonzero.

The following topics will be discussed:

1. Introduction to Hankel determinant; the original proof of APWW's theorem; Combinatorial proof; t-extensions of Hankel determinants.

2. Jacobi continued fractions and Hankel determinants; chopping method; grafting technique. A curious relation between the Thue-Morse sequence and the Golomb sequence. A formal power series F(x) such that the Hankel determinants of F(x) and F(x²) are equal.

3. Hankel continued fractions and Hankel determinants; automatic computer proof. It is well known that the continued fraction expansion of a quadratic irrational number is ultimately periodic; we prove a similar result for power series.