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Thursday, February 20, 2014

**Abstract:** In this talk I will present a polynomial analog of Stern's "diatomic" sequence, along with many of its properties. I will then define two subsequences of these polynomials. Various properties of these two interrelated subsequences are obtained; in particular, they have 0-1 coefficients and can be seen as extensions or analogues of the Fibonacci numbers. These sequences are then used to define two analytic functions as their respective limits. As an application we obtain evaluations of certain finite and infinite continued fractions whose partial quotients are doubly exponential. In a case of particular interest, the set of convergents has exactly two limit points. (This is joint work with K.B. Stolarsky). If time allows, I will also present joint work with John Cosgrave on mod $p^3$ analogs of binomial coefficient congurences due to Gauss and Jacobi, along with the related concept of a Gauss factorial.