function in the case of the small polaron approach. In Section III we present the procedure that provides the opportunity to avoid the iterative solution of the self-consistent equation for complex energies. The procedure is based on the continued fraction (CF) expansion of Green’s function and it gives a possibility to calculate consequently the coefficients of the CF expansion with the use of the quadrature method. In Section IV we offer the termination of the continued fraction which leads to the correct analytical properties of the resulting Green’s function. Numerical results for the self-energies and spectral functions, the relation of our results to the previous approaches and discussion are given in Section V. Section VI summarizes the results. An Appendix contains some details of the approach which gives the expression for integrals over the spectral density in terms of the chain representation of continuous fraction.

Part of our results were presented in Brief Reports [@bar1]. The paper presented here gives further results as well as the description of the new method and more details about the calculations.

Effective Hamiltonian and Small Polaron Green’s function
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Following Refs.[@em; @emrei; @bmui] we adopt the Hamiltonian that corresponds to one hole problem in 