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Aequationes mathematicae

, Volume 89, Issue 1, pp 187–206 | Cite as

d’Alembert’s other functional equation on monoids with an involution

  • Bruce Ebanks
  • Henrik Stetkær
Article

Abstract

This paper treats a functional equation of d’Alembert, namely f(x + y) − f(xy) = g(x)h(y), in the setting of groups and monoids that need not be abelian. Of the several possibilities for generalizing xy to the non-abelian case, here we replace it by σ(y)x where σ is an involutive automorphism, and find the complex-valued solutions (f, g, h) of f(xy) − f(σ(y)x) = g(x)h(y) in terms of multiplicative and additive functions.

Keywords

d’Alembert involutive automorphism functional equation non-abelian monoid 

Mathematics Subject Classification

39B32 39B52 

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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Mississippi State UniversityMississippi StateUSA
  2. 2.Aarhus UniversityAarhusDenmark

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