In recent years

with the development of several experimental techniques

for example

metal-organic chemical-vapour deposition

molecular beam epitaxy and electron beam lithography combined with reverse mesa etching

there has been of considerable interest in understanding of hydrogenicimpurity states in low-dimensional semiconductor heterostructures such as quantum wells

quantum wires and quantum dots. Recently

there has been great interest in investigating quantum dots both theoretically and experimentally. Due to the small structures of QD's

some physical properties such as optical and electron transport characteristic are quite different from those of the bulk materials. The study of the impurity states in these low dimensional structures is an important aspect to which many theoretical and experimental works based. Recent investi-(gations) in the reduced dimensionality show that for the impurity within the reduced dimensionality the impurity binding energy will be enhanced with the deduction of the dimensionality.In recent years

the problem of a bound magnetopolaron in a quantum dot has been extensively studied. Petukhov

et al

. studied the hopping and giant magnetoresisrance of a bound magnetic polaron in magnetic (semiconductors) and nanostrucrure with the golden rule at effective mass approximation. Charrour

et al

. presented a systematic study of the ground state binding energy of a hydrogenic impurity in cylindrical quantum dot subjected to an external strong magnetic field and performed calculations within the effective-mass approxi-(mation) using the variational procedure and considering an infinite confining potential on all surfaces of the (system.) Liu

et al

. investigated the properties of bound magnetopolaron in a quantum wires. Bouhassoune (

et al

.) studied magnetic field effect on the binding energy of a bound polaron in a quantum well by a variational method. Nguyen

et al

. analysized magnetic field effects of parabolic confining potentials and magnetic field on the binding energy of hydrogen impurites in quantum dots using a very simple trial function with only one (variational) parameter. Chen

et al

. derived ground state properties of the bound magnetopolaron using the linear combination operator and unitary transformation method. Corella

et al

. calculated the ground state and binding energy for the hydrogen impurity in a spherical quantum dot by using the variational method. Zhou

et al

. (derived) energy levels of the bound magnetopolaron by the variational method. Li

et al

. investigated the in-(fluence) of magnetic field on the binding energy of a hydrogenic impurity in cubic quantum dots

and so on. (However

) few investigators studied the average number of optical phonons around the electron in the quantum dot with the linear combination operator and a perturbation method so far. By using the linear combination operator and unitary transformation method

properties of the vibration frequency and the average number of optical phonons around the electron in the quantum dot are studied for the strong electron-LO-phonon coupling case. Results show that the vibration frequency decreases when the Coulomb potential enhances. While the vibration frequency and the average number of (optical) phonons decrease with the effective confinement length increasing and they increase with the electron-(LO- )phonon coupling strength.