A Levi–Civita Equation on Monoids, Two Ways

Annales Mathematicae Silesianae

article

Bruce Ebanks¹ 

[1] Department of Mathematics University of Louisville Louisville Kentucky 40292;Department of Mathematics and Statistics Mississippi State University Mississippi State MS 39762 USA

关键词: Levi–Civita equation;    sine addition formula;    cosine addition formula;    semigroup;    monoid;    exponential function;   

DOI  :  10.2478/amsil-2022-0009

学科分类：内科医学

来源: Walter de Gruyter GmbH

【 摘 要 】

We consider the Levi–Civita equationf(xy)=g1(x)h1(y)+g2(x)h2(y) f\left( {xy} \right) = {g_1}\left( x \right){h_1}\left( y \right) + {g_2}\left( x \right){h_2}\left( y \right) for unknown functions f, g1, g2, h1, h2 : S → ℂ, where S is a monoid. This functional equation contains as special cases many familiar functional equations, including the sine and cosine addition formulas. In a previous paper we solved this equation on groups and on monoids generated by their squares under the assumption that f is central. Here we solve the equation on monoids by two different methods. The first method is elementary and works on a general monoid, assuming only that the function f is central. The second way uses representation theory and assumes that the monoid is commutative. The solutions are found (in both cases) with the help of the recently obtained solution of the sine addition formula on semigroups. We also find the continuous solutions on topological monoids.

【 授权许可】

CC BY