On second differences in product form

Abstract

In this paper, we study the class of functions whose second difference admits product form. In particular, we determine the general solution of the functional equation ¶¶f (x1 yz, x2 y2) - f (x1 y1, x2 y2-1) - f (x1y1-1, x2 y2) + f (x1 y1-1, x2 y2-1) = g (x1, x2) h (y1,y2) (FE) ¶with f satisfying the conditions ¶¶f (x1 y1 z1, x2) = f (x1 z1 y1, x2) and f (x1,x2 y2 z2) = f (x1, x2 z2 y2)(KC)¶ for all \( x_i, y_i, z_i \in \bf {G}_i (i = 1,2) \), where G1 and G2 are arbitrary groups. Since we do not assume the groups to be abelian, the condition (KC) plays an important role for solving the (FE). The general solution of (FE) with (KC) is determined by deriving two related functional equations ¶¶g (x1 y1, x2y2) + g (x1 y1, x2 y2-1) + g (x1 y1-1, x2 y2) + g (x1 y1-1, x2 y2-1) = g (x1, x2) p (y1, y2) ¶ and ¶h (x1 y1, x2y2) - h (x1 y1, x2 y2-1) - h (x1 y1-1, x2 y2) + h (x1 y1-1, x2y1-1) = p (x1, x2) h (y1, y2). ¶The functional equation (FE)} generalizes a functional equation studied by J. Baker in 1969, McKiernan in 1972, and Haruki in 1980.