A functional equation on groups with involutions

Around the Sine Addition Law and d’Alembert’s Equation on Semigroups

“…Stetkaer [9] solved the variant f (xy) + f (σ(y)x) = 2f (x)f (y), x,y ∈ S, of d'Alembert's equation for unknown f : S → C, where σ : S → S is an involutive automorphism, showing that solutions have the form f = (m + m • σ)/2 for a multiplicative function m : S → C. Sabour et al [7] built on that result to solve f (xσ(y)) + f (τ (y)x) = 2f (x)g(y), x,y ∈ M, for f, g : M → C, where M is a group or a monoid generated by its squares. Motivated by that, Ng et al [6] found the solutions f, h, k : M → C of f (xασ(y)) + h(τ (y)x) = 2f (x)k(y), x,y ∈ M, for the case that M is a group and α ∈ M . Inspired by those results, the author [1] used the approach taken in [6] to solve (1) for the case that the monoid M is regular or generated by its squares.…”

“…Motivated by that, Ng et al [6] found the solutions f, h, k : M → C of f (xασ(y)) + h(τ (y)x) = 2f (x)k(y), x,y ∈ M, for the case that M is a group and α ∈ M . Inspired by those results, the author [1] used the approach taken in [6] to solve (1) for the case that the monoid M is regular or generated by its squares. In the present work we will solve (1) on a general monoid.…”

“…The method used in [6] proceeds from the starting point of the sine addition law f (xy) = f (x)g(y) + g(x)f (y) x, y ∈ S, (2) and we will start at the same place. The solution of (2) on a general semigroup was given recently by the author [2], then a simpler proof with a more detailed description of some of the solutions was given in [3].…”

“…Sections 4 and 5 are focused on solving (1). The path to the solution of that equation is shortened, relative to [6], by combining the solution of (2) with the result [10] by Stetkaer showing that the solutions f, g :…”

“…can be described in terms of multiplicative functions and solutions of the sine addition law. Returning to the path laid out in [6], we proceed to the solution of (1) on monoids. As is the case for (2), we find that on some monoids (contrary to the case for groups and monoids generated by their squares) there are solutions of (1) that cannot be completely described in terms of additive and multiplicative functions.…”