Occasion: Mathematics Colloquium
Place: University of South Alabama
AbstractApéry-like numbers are special integer sequences, going back to Beukers and Zagier, which are modelled after and share many of the properties of the numbers that underlie Apéry's proof of the irrationality of \(\zeta(3)\). Among their remarkable properties are connections with modular forms and so-called supercongruences, some of which remain conjectural. In the course of several examples, we demonstrate how these numbers and their connection with modular forms feature in various, apparently unrelated, problems. The examples are taken from personal research of the speaker and include the theories of short random walks, binomial congruences, series for \(1/\pi\), and positivity of rational functions. Finally, we return to the discussion of supercongruences and report on new perspectives and recent progress.
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