As the author writes in the Introduction, “... some classical ideas in the theory of functional equations can be used successfully to solve different types of functional equations and systems of functional equations on hypergroups but the way is far from formal generalization. .... We consider the sine and cosine functional equations (1) and (2). It may be of some interest to characterize some ‘trigonometric type’ or ‘hyperbolic type’ functions on different hypergroups. The aim of this work is to present some new methods and results related to this problem.” – The equation labels (1) and (2) are mentioned repeatedly on pages 225 and 226. It may be (would have been) helpful if the reader had been directed to pages 228 and 230, respectively, where these equations are presented for the first time in this paper: \(\:(1)\; f(n\ast m)=f(n)g(m)+f(m)g(n),\quad(2)\;f(n\ast m)=f(n)f(m)-g(m)g(n)\,.\;\) Here \((\mathbb{N},\ast)\) is the polynomial hypergroup associated to the sequence of polynomials \(\{P_n\}\) satisfying \(P_0(x)=1,\, P_1(x)=x,\;P_n(x)=a_n P_{n-1}(x)+b_n P_n(x)+c_nP_{n+1}(x)\;(x\in\mathbb{R};a_{n+1}>0, b_n\geq 0, c_n>0,\, a_n+b_n+c_n=1;\; 1\leq n\in\mathbb{N}).\) The author offers the general solutions \(f,g:\mathbb{N}\to\mathbb{C}\;(f\,\) not identically \(0\)) of each of these two equations satisfied for all \(m,n\in\mathbb{N}.\) The proofs are based on spectral analysis and spectral synthesis. The paper ends with “Functional equations of similar type on polynomial hypergroups in several variables will be considered elsewhere”.