D\u2019Alembert\u2019s and Wilson\u2019s equations on semigroups

Abstract

In this paper we consider a generalization of d’Alembert’s equation and Wilson’s equation on commutative semigroups using only the semigroup operation, ie. we consider the functional equation h(x+2y)+h(x)=2f(y)h(x+y),x,y∈S,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}&h(x+2y)+h(x)=2f(y)h(x+y),\\ x,y\\in S, \\end{aligned}$$\\end{document}where f,h:Srightarrow mathbb {K}, (S,+) is a commutative semigroup, mathbb {K} is a quadratically closed field, text {char},mathbb {K}ne 2.