Some curious infinite products

Abstract: 

We will discuss some sequences whose generating functions are infinite products which act reasonably under the Frobenius operator, $x\mapsto x^k$, for some $k$. Such sequences, e.g. the Thue-Morse sequence and Stern’s diatomic sequence, have curious behaviour. In this talk, I will look at these function from two perspectives, the first about asymptotics at ’special’ points, and then about asymptotics 'almost everywhere’.