Sine and cosine type functional equations on hypergroups.

Let (X , *) {(X,*)} be a hypergroup and let w 0 {w_{0}} be a fixed measure on X. In this paper we study the two functional equations 〈 δ x * δ y * ω 0 , g 〉 + 〈 δ x * δ y ˇ * ω 0 , g 〉 = 2 ⁢ g ⁢ (x) ⁢ g ⁢ (y) , x , y ∈ X , \langle\delta_{x}*\delta_{y}*\omega_{0},g\rangle+\langle\delta_{x}*\delta_{% \check{y}}*\omega_{0},g\rangle=2g(x)g(y),\quad x,y\in X, and 〈 δ x * δ y ˇ * ω 0 , f 〉 - 〈 δ x * δ y * ω 0 , f 〉 = 2 ⁢ f ⁢ (x) ⁢ f ⁢ (y) , x , y ∈ X , \langle\delta_{x}*\delta_{\check{y}}*\omega_{0},f\rangle-\langle\delta_{x}*% \delta_{y}*\omega_{0},f\rangle=2f(x)f(y),\quad x,y\in X, where g , f : X → ℂ {g,f:X\to\mathbb{C}} are continuous and bounded functions to be determined. We express the solutions of the two functional equations in terms of multiplicative maps on (X , *) {(X,*)}. As an application we give the solution of the two functional equations on polynomial and Sturm–Liouville hypergroups. [ABSTRACT FROM AUTHOR]