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Explore advanced mathematical concepts in this 47-minute lecture from Institut des Hautes Etudes Scientifiques (IHES), where Karol Penson delves into positive integer sequences expressible through ratios of factorial products and Gamma functions. Learn how generating functions of these sequences become generalized hypergeometric functions with algebraic properties, building upon the foundational work of Beukers-Heckman and Bober. Discover new families of shifted binomials and sequences derived from recent integrality criteria, complete with closed-form parametric expressions for their algebraic equations. Examine how these sequences can be interpreted as power moments of weight functions with finite support, and understand the relationship between their algebraic equations and weight solutions in the Hausdorff moment problem through inverse Mellin transform. The lecture, presented in collaboration with G. H. E. Duchamp, M. Kontsevich, and G. Koshevoy, advances the field beyond existing classifications while demonstrating explicit correspondences, particularly in cases involving elementary functions in radical form.
