Wednesday, July 22, 2020 — 4:30 PM EDT

Adam Humeniuk, Department of Pure Mathematics, University of Waterloo

"Generatingfunctionology: basics and approximation"

A generating function is a device for studying a sequence by trapping it in the coefficients of a power series. I'll give a brief crash course on "generatingfunctionology", and show you how to write down the generating function of Fibonacci numbers. This gives, for instance, an exact formula for the nth Fibonacci number. We don’t usually care whether the series converges, and work in the setting of “formal” power series.

But! By asking the question of where the power series converges, we get extra information. Using only some gentle complex analysis, we can get simple asymptotic formulae for the sequence we started with. We’ll figure out the generating function for the ordered Bell numbers. (These count orderings where ties are allowed, like the outcomes of a race.) Examining the poles where the generating function blows up will reveal a remarkably simple approximate formula for the nth ordered Bell number.

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