Two general functional equations

TL;DR: In this paper, it was shown that the functions of equations (1) and (2) are expressible in terms of the functions <p(x) and ip(x), which satisfy the Cauchy equations (3), given above as special cases of (1), and the results of this paper are of sufficient generality to permit immediate

Abstract: in which x and y are independent variables and ƒ(#), g(x), h(x), ft (ad, F(x), Gf(x), H(x), K{x) are functions to be determined. Special cases of equations (1) and (2) have been discussed in the literature. Some of the more familiar special cases are h(x) = Jc(x) = ƒ(#) = ip(x), g(x) = 0, gix) = k(x) =f{x\ h(x) = 2f(x), G(x) = 1, H{x) = Kix) = F(x) = 9(x), G(x) = F(x), H{x) EEE F\x\ K(x) = F\x) — 1, G(x) = Fix), Hix) = F\x\ K(x) =—F(x). In this paper no relationships are assumed between the functions in equation (1) or the functions in equation (2). Furthermore, no restrictions (such as continuity, differentiability, etc.) are imposed on the functions. The variables, x and y, are not assumed real nor must they necessarily be complex. The author shows that the functions of equations (1) and (2) are expressible in terms of the functions <p(x) and ip(x) which satisfy the Cauchy equations (3) <p(x-\~y) = sp(#) + y(y), (4) V>(x + y) = ip(x)tp(y), given above as special cases of (1) and (2). The results of the paper are of sufficient generality to permit immediate