Summary Functional equations of the form$$f_1 (x + y)f_2 (x - y) = \sum\limits_{i = 1}^n {\alpha _i (x)\beta _i (y)} $$ are investigated on R and on more general domains. The paper is divided into five sections. After two introductory sections we solve in Section 3 two special cases of the equation$$f_1 (x + y)f_2 (x -
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