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In the paper we find solution representations in the compact integral form tothe Cauchy problem for a general form of the Euler--Poisson--Darboux equationwith Bessel operators via generalized translation and spherical mean operatorsfor all values of the parameter $k$, including also not studying beforeexceptional odd negative values. We use a Hankel transform method to proveresults in a unified way. Under additional conditions we prove that adistributional solution is a classical one too. A transmutation property forconnected generalized spherical mean is proved and importance of applyingtransmutation methods for differential equations with Bessel operators isemphasized. The paper also contains a short historical introduction ondifferential equations with Bessel operators and a rather detailed referencelist of monographs and papers on mathematical theory and applications of thisclass of differential equations.

The general form of the Euler--Poisson--Darboux equation and application of transmutation method