Some identities of the sums-of-tails type

TL;DR

This paper introduces new sums-of-tails identities involving parameters, generalizes Ramanujan's identities, and derives new representations for series related to mock theta functions, with applications to combinatorics and existing identities.

Contribution

It presents a novel sums-of-tails identity with parameters, generalizes Ramanujan's identity, and provides new series representations for mock theta functions.

Findings

A new sums-of-tails identity involving parameters b and d.

Ramanujan's sums-of-tails identity is a special case of the new result.

Two new series representations for a mock theta function.

Abstract

A new sums-of-tails identity involving two parameters $b$ and $d$ is obtained and is used to derive more results of similar type. One of Ramanujan's sums-of-tails identities from the Lost Notebook is shown to be a special case of our result. In the course of deriving Ramanujan's identity, we obtain a new result of combinatorial significance. Two new representations for an infinite series associated to a mock theta function are derived. Also, we give an application of an identity of Andrews and Onofri.