Fractional squeezing-Hankel transform based on the induced entangled state representations

Based on the fact that the quantum mechanical version of Hankel transform kernel (the Bessel function) is just the transform between |q,r> angle 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(a₁†a₂†+a₁a₂)}e^{-iπa₂†a₂}, which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved.