2015/12/09 by Bouikhalene Belaid, Belaid, Bouikhalene, Elqorachi Elhoucien +1 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #math.CA #math.NT
paper · pdf · doi:10.48550/arxiv.1512.06753
9 pages
arxiv created 2015/12/09 · arxiv updated 2015/12/22
In \citeSt3 H. Stetkær obtained the solutions of Van Vleck's functional equation for the sine f(xτ(y)z0)-f(xyz0) =2f(x)f(y), x,y∈ G, where G is a semigroup, τ is an involution of G and z0 is a fixed element in the center of G. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck's functional equation for the sine μ(y)f(xτ(y)z0)-f(xyz0) =2f(x)f(y), x,y∈ G, where μ : G\longrightarrow ℂ is a multiplicative function such that μ(xτ(x))=1 for all x∈ G. Furthermore, we obtain the solutions of a variant of Van Vleck's functional equation for the sine μ(y)f(σ(y)xz0)-f(xyz0) = 2f(x)f(y), x,y∈ G on monoids, and where σ is an automorphism involutive of G.