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On trigonometric functional equations of rectangular type

Research paper by Jukang Chung, Palaniappan Kannappan, Prasanna K. Sahoo

Indexed on: 01 Feb '97Published on: 01 Feb '97Published in: Aequationes mathematicae



Abstract

Letf, G1 × G2 → C, where Gi (i = 1, 2) denote arbitrary groups and C denotes the set of complex numbers. The general solutions of the following functional equationsf(x1y1,x2y2) +f(x1y1,x2y2-1) +f(x1y1-1,x2y2) +f(x1y1-1,x2y2-1) =f(x1,x2)F(y1,y2) +F(x1,x2)f(y1,y2) (1) andf(x1y1,x2y2) +f(x1y1,x2y2-1) +f(x1y1-1,x2y2) +f(x1y1-1,x2y2-1) =f(x1,x2)f(y1,y2) +F(x1,x2)F(y1,y2) (2) are determined assuming thatf satisfies the conditionf(x1y1z1, x2) = f(x1z1y1, x2), f(x1, x2y2z2) = f(x1, x2z2y2) (C) for allxi, yi, xi ∈ Gi (i = 1, 2). The functional equations (1) and (2) are generalizations of the well known rectangular type functional equationf(x1 + y1, x2 + y2) + f(x1 + y1, x2 − y2) + f(x1 − y1, x2 + y2) + f(x1 − y1, x2 − y2) = 4f(x1, x2) studied by J. Aczel, H. Haruki, M. A. McKiernan and G. N. Sakovic in 1968.