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.