We solve the functional equation $$f(xy) + g(\sigma (y)x) = h(x)k(y)$$ f ( x y ) + g ( σ ( y ) x ) = h ( x ) k ( y ) for complex-valued functions f , g , h , k on groups or monoids generated by their squares, where $$\sigma $$ σ is an involutive automorphism. This contains both classical d’Alembert equations $$g(x + y) + g(x - y) = 2g(x)g(y)$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) g ( y ) and $$f(x + y) - f(x - y) = g(x)h(y)$$ f ( x + y ) - f ( x - y ) = g ( x ) h ( y ) in the abelian case, but we do not suppose our groups or monoids are abelian. We also find the continuous solutions on topological groups and monoids.

中文翻译：

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幺半群上 d'Alembert 泛函方程的完全 Pexiderized 变体

我们求解函数方程 $$f(xy) + g(\sigma (y)x) = h(x)k(y)$$ f ( xy ) + g ( σ ( y ) x ) = h ( x ) k ( y ) 对于复值函数 f 、 g 、 h 、 k 由它们的平方生成的群或幺半群，其中 $$\sigma $$ σ 是对合自同构。这包含两个经典的 d'Alembert 方程 $$g(x + y) + g(x - y) = 2g(x)g(y)$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) g ( y ) 和 $$f(x + y) - f(x - y) = g(x)h(y)$$ f ( x + y ) - f ( x - y ) = g ( x ) h ( y ) 在阿贝尔情况下，但我们不假设我们的群或幺半群是阿贝尔的。我们还找到了拓扑群和幺半群的连续解。