Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between |q, r⟩ and (s, r ′ |, two induced entangled state representations are given, and working with them we derive fractional squeezing-Hankel transform (FrSHT) caused by the operator e −iα(a † 1 a † 2 +a 1 a 2 ) e −iπa † 2 a 2 , which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved.

Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between [Formula: see text] and [Formula: see text], two induced entangled state representations, and working with them, we derive fractional Fourier–Hankel transformation (FrFHT) caused by the operator [Formula: see text], where [Formula: see text] is named the core operator and is essential to the fractional transformation. The fractional property (additive rule) of the FrFHT can be explicitly proved.