A relation between the Cauchy wavelets and the step-up/down operator of a kind of orthogonal wavepacket system

It is well known that the over-complete Cauchy wavelet system with continuous parameters is the eigenfunction system of the operator T+kJ (T:multiplication by t, J:integral op.). This is quite parallel to the well-known relation between the coherent state system and the boson annihilation operator used in quantum mechanics, in which the annihilation operator is the step-down operator of the number states. In this paper, for the Cauchy wavelet system, we show a similar relation to this on the step-up/down operator of a kind of orthogonal function system. The operator T+kJ itself is not the step-down operator but a kind of rational function of this operator,and is the step-down operator of a kind of orthogonal wavepacket system which is the eigenfunction system of the analogue of `number´ operator