Series with general exponents

Abstract

We define the Artinian and Noetherian algebra which consist of formal series involving exponents which are not necessarily integers. All of the usual operations are defined here and characterized. As an application, we compute the algebra of symmetric functions with nonnegative real exponents. The applications to logarithmic series and the Umbral calculus are deferred to another paper. On d\'efinit ici les alg\`ebres Artinienne et Noetherienne comme \'etant des alg\`ebres constitu\'ees des s\'eries formelles \`a exposants pas n\'ecessairement entiers. On definit sur ces alg\`ebres toutes les op\'erations classiques et on les caracterise. Comme exemple d'exploitation de cette th\'eorie, on s'interesse \`a alg\`ebre de fonctions sym\'etriques &agrave exponsants r\`eels en nonn\'egatifs. Une autre publication est consacr\'ee aux applications aux series logarithmiques et au calcul ombral.