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New families of $E$-functions are described in the context of the compactsimple Lie groups O(5) and G(2). These functions of two real variablesgeneralize the common exponential functions and for each group, only one familyis currently found in the literature. All the families are fully characterized,their most important properties are described, namely their continuous anddiscrete orthogonalities and decompositions of their products.

Six types of $E-$functions of the Lie groups O(5) and G(2)