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Abstract: Of the many remarkable formulae of Ramanujan, the Rogers-Ramanujan identities, relating two generating functions in an unexpected way, are probably the most famous. They are interesting to number-theorists, algebraists, analysts, and physicists alike, and they have the touch of the extraordinary because of the mysterious appearance of the number 5. The usual analytical arguments are quite involved, and the combinatorial bijective proof is extremely complicated. We start with an unusual formula for the Fibonacci numbers that comes from counting certain lattice paths, and show that the Rogers-Ramanujan identities are just the weighted versions of the Fibonacci case. The combinatorial tools that are used on the way are inclusion-exclusion, binomial and Gauss coefficients, number partitions, and the triple product theorem of Jacobi.
Colloquium - 16:00