Hausdorff Moment Problems for Combinatorial Numbers: Heuristics via Meijer Functions

Explore a 45-minute lecture on Hausdorff moment problems for combinatorial numbers, focusing on integral ratios of factorials as power moments. Delve into solving moment problems by deriving exact expressions for weight functions W(x) on the support (0, R). Examine various combinatorial sequences, including Tutte's planar map enumerations, Fuss-Catalan and Raney numbers, constellation numbers, and ratios of multiple factorials. Investigate Bober ratios, Rodriguez Villegas sequences, and their algebraic ordinary generating functions. Discover a persistent pattern relating Meijer G-encodings of generating functions G(z) and weight functions W(x). Explore the potential for automating solutions using Meijer G-functions and discuss limitations and open problems in this approach. Learn from collaborative insights with N. Behr, G. H. E. Duchamp, K. Górska, M. Kontsevich, and G. Koshevoy in this advanced mathematical exploration presented by Karol Penson from LPTMC, Sorbonne Université Paris.