Title: The Stern Sequence and Stern Polynomials
Abstract: In 1858, responding to a question or Eisenstein about denominators in Farey sequences, Stern defined a sequence s(n) by s(0) = 0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1) and proved (decades before Cantor!) that every positive rational occurs exactly once as s(n)/s(n+1). One generalization is the Stern polynomial: s(0,x) = 0, s(1,x) = 1, s(2n,x) = xs(n,x), s(2n+1,x) = s(n,x) + s(n+1,x). We'll discuss some of the amazing properties of {s(n,x)}, especially factorization and the location of zeros.