Abstract
We solve the functional equation \(f(xy) + g(\sigma (y)x) = h(x)k(y)\) for complex-valued functions f, g, h, k on groups or monoids generated by their squares, where \(\sigma \) is an involutive automorphism. This contains both classical d’Alembert equations \(g(x + y) + g(x - y) = 2g(x)g(y)\) and \(f(x + y) - f(x - y) = g(x)h(y)\) 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.
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19 March 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00025-021-01351-3
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Ebanks, B. A Fully Pexiderized Variant of d’Alembert’s Functional Equations on Monoids. Results Math 76, 17 (2021). https://doi.org/10.1007/s00025-020-01335-9
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DOI: https://doi.org/10.1007/s00025-020-01335-9