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This is structurally identical to the property of [[exponentiation]] that {{math|''a''<sup>''m''</sup>''a''<sup>''n''</sup> {{=}} ''a''<sup>''m''+''n''</sup>}}, i.e. the special case {{math|''f''(''x''){{=}}''ax''}}.
In general, for arbitrary general (negative, non-integer, etc.) indices {{mvar|m}} and {{mvar|n}}, this relation is called the '''translation functional equation''', cf. [[Schröder's equation]] and [[Abel equation]]. On a logarithmic scale, this reduces to the '''nesting property''' of [[Chebyshev polynomials]], {{math|''T''<sub>''m''</sub>(''T''<sub>''n''</sub>(''x'')){{=}}''T''<sub>''m n''</sub>(''x'')}}, since {{math|''T''<sub>''n''</sub>(''x'') {{=}} cos(''n'' arcos(''x'' ))}}.
The relation {{math|(''f''<sup> ''m''</sup> )<sup>''n''</sup>(''x'') {{=}} (''f''<sup> ''n''</sup> )<sup>''m''</sup>(''x'') {{=}} ''f''<sup> ''mn''</sup>(''x'')}} also holds, analogous to the property of exponentiation that {{math|(''a''<sup>''m''</sup> )<sup>''n''</sup> {{=}}(''a''<sup>''n''</sup> )<sup>''m''</sup> {{=}} ''a''<sup>''mn''</sup>}}.
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