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〉 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 $e^{-i\alpha (a_1^{†}{a}_2^{†}+a_1{a}_2)}{e}^{-i\pi a_2^{†}{a}_2}$, which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved. (paper)