On sum-of-tails identities

In this article, a finite analogue of the generalized sum-of-tails identity of Andrews and Freitas is obtained. We derive several interesting results as special cases of this analogue, in particular, a recent identity of Dixit, Eyyunni, Maji and Sood. We derive a new extension of Abel's lemma with the help of which we obtain a one-parameter generalization of a sum-of-tails identity of Andrews, Garvan and Liang, an identity of Ramanujan as well as two new results - one for Ramanujan's function σ(q) and another for the function recently introduced by Andrews and Ballantine. Later we introduce a new generalization FFW<inf>c</inf>(n) of a function of Fokkink, Fokkink and Wang and derive an identity for its generating function. This gives, as a special case, a recent representation for the generating function of spt(n) given by Andrews, Garvan and Liang. We also obtain some weighted partition identities along with new representations for two of Ramanujan's third order mock theta functions through combinatorial techniques.