Solution for 3rd order Cauchy Difference Equation in Free Monoid

Abstract

— Let k: C ? D  be a function, where (C,.) is a Group and (D,+) is an Abelian Group. In  this article, the Third Order Cauchy difference of k:E⁽³⁾  k (c_1,c_2,c_3,c₄)= k(c₁c₂c₃c₄) –  k(c₁c₂c₃)  – k(c₁c₃c₄) – k (c₁c₂c₃c₄) –k (c₂c₃c₄) +  k(c₁c₂) +  k (c₁c₃) +k(c₁c₄)+k(c₂c₃)+ k(c₂c₄)+k(c₃c₄) –k(c₁) – k(c₂) –k(c₃) –k(c₄)   ? c_1,c_2,c_3,c₄ C    is studied. We give solutions of E⁽³⁾k=0  on Free Monoid.