Unilateral Generalized Lambert Series Identities

Lambert series are widely used in analytic number theory,hypergeometric series,combinatorics,elliptic functions,theta functions.In this paper,firstly,we prove three unilateral generalized Lambert series identities by using the partial fraction decomposition theorem of rational function and the method of calculating residue,we can look upon these identities as the generalized results of Andrews,Lewis,Liu.Then,we obtained the corresponding Lambert series identities as their applications from the three identities.which are different from the results of Andrews,et al.