On the dual nature of partial theta functions and Appell-Lerch sums

In recent work, Hickerson and the author demonstrated that it is useful to think of Appell--Lerch sums as partial theta functions. This notion can be used to relate identities involving partial theta functions with identities involving Appell--Lerch sums. In this sense, Appell--Lerch sums and partial theta functions appear to be dual to each other. This duality theory is not unlike that found by Andrews between various sets of identities of Rogers-Ramanujan type with respect to Baxter's solution to the hard hexagon model of statistical mechanics. As an application we construct bilateral $q$-series with mixed mock modular behaviour.Comment: To be published in Advances in Mathematic

The generalized Borwein conjecture. II. Refined q-trinomial coefficients

Transformation formulas for four-parameter refinements of the q-trinomial coefficients are proven. The iterative nature of these transformations allows for the easy derivation of several infinite series of q-trinomial identities, and can be applied to prove many instances of Bressoud's generalized Borwein conjecture.Comment: 36 pages, AMS-LaTe

Two truncated identities of Gauss

Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.Comment: 9 pages, final versio

Static and dynamic modifications to photon absorption:The effects of surrounding chromophores

This Letter investigates the influence, on the molecular absorption of light, of surrounding chromophores. Two novel rate contributions are identified - one vanishing for a medium with no static dipole moment. The other, dynamic term is used to model a system of primary absorbers and secondary chromophores distributed in a host medium. Further modification provides a basis for modelling a case where the medium is, itself, marginally absorptive, thus accounting for optical losses as the input propagates through the surrounding host. The results facilitate tailoring of secondary chromophore and host effects in the pursuit of materials with specific absorption features