location:  Publications → journals
Search results

Search: MSC category 33C05 ( Classical hypergeometric functions, ${}_2F_1$ )

 Expand all        Collapse all Results 1 - 2 of 2

1. CJM Online first

Ovchinnikov, Alexey; Wibmer, Michael
 Tannakian categories with semigroup actions Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications. Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parametersCategories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06

2. CJM 2000 (vol 52 pp. 31)

Chan, Heng Huat; Liaw, Wen-Chin
 On Russell-Type Modular Equations In this paper, we revisit Russell-type modular equations, a collection of modular equations first studied systematically by R.~Russell in 1887. We give a proof of Russell's main theorem and indicate the relations between such equations and the constructions of Hilbert class fields of imaginary quadratic fields. Motivated by Russell's theorem, we state and prove its cubic analogue which allows us to construct Russell-type modular equations in the theory of signature~$3$. Categories:33D10, 33C05, 11F11
 top of page | contact us | privacy | site map |