(Japanese / 日本語)
(Japanese / 日本語)
Speaker: Frits Beukers (Utrecht University)
Date: 9th April 2022 (Sat) from 16:00 JST
Title: Strange evaluation of hypergeometric series
Abstract: Hypergeometric series form an abundantly rich source of identities and unexpected evaluations. It is also a testing ground for phenomena in irrationality theory. In this presentation we shall discuss evaluations which are perhaps less well-known. Namely products of values of Gamma-functions at rational arguments, often called 'strange evaluations'. This is a joint project with Jens Forsgard which was originally inspired by work of Akihito Ebisu in AMS memoir 1177.
Speaker: Hajime Kaneko (University of Tsukuba)
Date: 14th May 2022 (Sat) from 16:00 JST
Title: Multiplicative analogue of the Lagrange spectrum related to linear recurrences
Abstract: Fractional parts of geometric progressions and more general linear recurrences have been long investigated. However, little is known on the fractional parts of such sequences. For instance, it is still unknown whether the fractional parts of $(3/2)^n$ ($n=0,1,\ldots$) are dense in the interval $[0,1]$. In this talk, we introduce new numerical systems, different from the beta expansion, to study the fractional parts of linear recurrences. As applications, we investigate the topological properties on the set of the maximum limit points of fractional parts. In particular, we obtained multiplicative analogue of the Lagrange spectrum. This is a joint work with Shigeki Akiyama and Teturo Kamae.
Speaker: Akihito Ebisu (Chiba Institute of Technology)
Date: 11th June 2022 (Sat) from 17:00 JST
Title: Identities for hypergeometric functions: from the view point of contiguous relations
Abstract: There are many identities for hypergeometric functions (HGF): Summation formulas (often called "Strange evaluations"), Transformation formulas, Quadratic relations, Continued fraction expansions of HGF. Some of those appear in Number theory and related topics. In this talk, we discuss such identities from the view point of contiguous relations for HGF.
Speaker: Anthony Poëls (JSPS research fellow at Nihon University)
Date: 9th July 2022 (Sat) from 16:00 JST
Title: Padé approximation for a class of hypergeometric functions
Abstract: This is a joint work with Makoto Kawashima. We obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. In this talk, we will construct explicitly Padé approximations by using a formal method and show that the associated sequences satisfy a specific Poincaré-type recurrence, which leads us to the expected irrationality measures.
Speaker: Hitoshi Nakada (Keio University)
Date: 10th September 2022 (Sat) from 16:00 JST
Title: Metric theory of complex continued fractions -- dynamic point of view
Abstract: We start with the history of the metric theory of continued fractions. Then we consider the notion of the natural extension of a non-invertible measure preserving transformation and give a representation of the natural extension of the real continued fraction map as a planar map. Main point of this representation is that there is a nice relation between the domain of the planar map and a set of geodesics over the upper-half plane in the case of real numbers. Finally we apply the method developed in the real case to the complex case. We consider some complex continued fraction maps and show how the method works and how it doesn't. We note that one can define a representation of the natural extension of a complex continued fraction map on a set of geodesics on the upper-half space.
Speaker: Miyazaki takafumi (Gunma University)
Date: 8th October 2022 (Sat) from 16:00 JST
Title: Number of solutions to a special type of unit equations in two unknowns II
Abstract: This talk is a continuation of the one given by I. Pink (University of Debrecen) on April 17, 2021. The topic is the best possible general estimate of the number of solutions to a special type of the unit equations in two unknowns over the rationals. R. Scott and R. Styer conjectured in 2016 that for any fixed relatively prime integers a,b and c greater than 1 the equation a^x+b^y=c^z has at most one solution in positive integers x,y and z, except for specific cases. In this talk we give a brief introduction on the conjecture, and present our results with their proofs, which in particular provides an analytic proof of the celebrated theorem of Scott (1993) solving the conjecture for c=2 in a purely algebraic manner. This is a joint work with István Pink.
Speaker: Anup Dixit (Institute of Mathematical Sciences, Chennai, India)
Date: 12th November 2022 (Sat) from 16:00 JST
Title: On generalized Diophantine m-tuples
Abstract: A set of positive integers {a_1, a_2, ... , a_m} is said to be a Diophantine m-tuple if a_i a_j +1 is a perfect square for all distinct i and j. A natural question is how large can a Diophantine tuple be. This problem has been studied for over millennia, starting from Diophantus to Fermat to Euler etc. In this context, the folklore Diophantine quintuple conjecture, recently settled by He, Togbe and Ziegler, states that there are no Diophantine quintuples. In this talk, we will discuss a generalization of this problem to higher powers. This is recent joint work with Saunak Bhattacharjee and Dishant Saikia, building on previous joint work with Ram Murty and Seoyoung Kim.