Séminaire du 22 novembre 2010, 10h30: Mark
Daniel Ward, Purdue University.
Solution of a Wilf Problem, and Asymptotic Analysis of the Nörlund Polynomials
We present the recent solution of the third of Herb Wilf's eight
Unsolved Problems, about asymptotic rational approximations to Pi. We
provide a complete asymptotic analysis of the coefficients of Wilf's
generating function. The asymptotic expansion is naturally additive
(not multiplicative); each term of the expansion contains oscillations,
which we precisely characterize. We also discuss new results on the
asymptotic growth of the Nörlund polynomials evaluated at any fixed
positive real number. In both problems, our methods give an asymptotic
expansion to any desired degree of accuracy. The methods of analysis
use analytic combinatorics in the style of Flajolet and Sedgewick. In
particular, the proofs utilize singularity analysis, Hankel contours,
and transfer theorems.
Last modified: Mon Sep 20 14:24:06 CEST 2010