(English / 英語)
(English / 英語)
講演者: Frits Beukers (Utrecht University)
日時: 2022年4月9日(土)16:00から
タイトル: Strange evaluation of hypergeometric series
アブストラクト: 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.
講演者: 金子 元 (筑波大学)
日時: 2022年5月14日(土)16:00から
タイトル: Multiplicative analogue of the Lagrange spectrum related to linear recurrences
アブストラクト: 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.
講演者: 蛭子彰仁 (千葉工業大学)
日時: 2022年6月11日(土)17:00から(いつもとは異なる開始時間です)
タイトル: Identities for hypergeometric functions: from the view point of contiguous relations
アブストラクト: 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.
数論に現れる超幾何関数に関する恒等式を幾つか復習した後、 それらが超幾何関数の隣接関係式の 文脈の中で導かれることを話したい。
講演者: Anthony Poëls (JSPS research fellow at Nihon University)
日時: 2022年7月9日(土)16:00から
タイトル: Padé approximation for a class of hypergeometric functions
アブストラクト: 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.
講演者: 仲田 均 (慶應義塾大学)
日時: 2022年 9月10日 (土) 16:00から
タイトル: Metric theory of complex continued fractions -- dynamic point of view
アブストラクト: 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.
講演者: 宮崎隆史 (群馬大学)
日時: 2022年10月8日(土)16:00から
タイトル: Number of solutions to a special type of unit equations in two unknowns II
アブストラクト: TThis 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.
講演者: Anup Dixit (Institute of Mathematical Sciences, Chennai, India)
日時: 2022年11月12日(土)16:00から
タイトル: On generalized Diophantine m-tuples
アブストラクト: 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.