A joint generalization of Van Vleck's and Kannappan's equations on groups

Brahim Fadli; Driss Zeglami; Samir Kabbaj

doi:10.1515/apam-2015-0026

Published by De Gruyter June 16, 2015

A joint generalization of Van Vleck's and Kannappan's equations on groups

Brahim Fadli , Driss Zeglami and Samir Kabbaj

From the journal Advances in Pure and Applied Mathematics

https://doi.org/10.1515/apam-2015-0026

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Abstract

Let G be a group, and let σ be an involutive automorphism on G. We determine the complex-valued solutions (F1,F2,f) of the functional equation F1(xy)+F2(σ(y)x)=f(x)f(y) (x,y∈G) in terms of characters and additive functions. From this we find the complex-valued solutions (f,g) of the equation f(σ(y)xz0)+g(xyz0)=2f(x)f(y), where z₀ is a fixed element in G. This equation provides a joint generalization of many functional equations such as d'Alembert's, Kannappan's or Van Vleck's equations.

Keywords: Van Vleck's equation; Kannappan's equation; involutive automorphism; group character

MSC: 39B32; 39B52

We wish to express our thanks to the referees for useful comments.

Received: 2015-4-28

Revised: 2015-6-2

Accepted: 2015-6-5

Published Online: 2015-6-16

Published in Print: 2015-7-1

A joint generalization of Van Vleck's and Kannappan's equations on groups

Abstract

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