A double-sum Kronecker-type identity

We prove a double-sum analog of an identity known to Kronecker and then express it in terms of functions studied by Appell and Kronecker's student Lerch, in so doing we show that the double-sum analog is of mixed mock modular form. We also give related symmetric generalizations.Comment: Major revisions. Identities (1.10) and (1.11) are ne

Mechanism of the Enzymic Reduction of N_2: The Binding of Adenosine 5'-Triphosphate and Cyanide to the N_2-reducing System

The in vitro reduction of N_2 is a complex process involving at least six different reactants: two proteins [1,2] for which the names azoferredoxin (AzoFd) and molybdoferredoxin (MoFd) have been proposed[3], an electron source, the electron acceptor, ATP[4], and Mg2+[5-7]. One of the goals of research in this area is to define the orderly and quantitative participation of these reactants leading to the reduction of the electron acceptor with concomitant breakdown of ATP to ADP and inorganic phosphate[7]. The work described in this paper shows that (1) AzoFd reversibly binds both ATP, a reactant in N2 reduction, and ADP, a specific inhibitor of N2 reduction, and (2) MoFd reversibly binds cyanide, which is also reduced by the N_2-reducing system. It is suggested that the binding of ATP and of cyanide are partial reactions of the N_2-reducing system

A general formula for Hecke-type false theta functions

In recent work where Matsusaka generalizes the relationship between Habiro-type series and false theta functions after Hikami, five families of Hecke-type double-sums of the form \begin{equation*} \left( \sum_{r,s\ge 0 }-\sum_{r,s<0}\right)(-1)^{r+s}x^ry^sq^{a\binom{r}{2}+brs+c\binom{s}{2}}, \end{equation*} where $b^2-ac<0$, are decomposed into sums of products of theta functions and false theta functions. Here we obtain a general formula for such double-sums in terms of theta functions and false theta functions, which subsumes the decompositions of Matsusaka. Our general formula is similar in structure to the case $b^2-ac>0$, where Mortenson and Zwegers obtain a decomposition in terms of Appell functions and theta functions.Comment: The number of pages is perfect. The title has change

Three new identities for the sixth-order mock theta functions

Ramanujan's lost notebook contains many mock theta functions and mock theta function identities not mentioned in his last letter to Hardy. For example, we find the four tenth-order mock theta functions and their six identities. The six identities themselves are of a spectacular nature and were first proved by Choi. We also find eight sixth-order mock theta functions in the lost notebook, but among their many identities there is only a single relationship like those of the tenth-orders. Using Appell function properties of Hickerson and Mortenson, we discover and prove three new identities for the sixth-order mock theta functions that are in the spirit of the six tenth-order identities. We also include an additional nineteen tenth-order-like identities for various combinations of second, sixth, and eighth-order mock theta functions

On Hecke-type double-sums and general string functions for the affine Lie algebra A1(1)A_{1}^{(1)}

We demonstrate how formulas that express Hecke-type double-sums in terms of theta functions and Appell--Lerch functions -- the building blocks of Ramanujan's mock theta functions -- can be used to give general string function formulas for the affine Lie algebra $A_{1}^{(1)}$ for levels $N=1,2,3,4$.Comment: 27 pages. arXiv admin note: text overlap with arXiv:2107.0622

A heuristic guide to evaluating triple-sums

Using a heuristic that relates Appell--Lerch functions to divergent partial theta functions one can expand Hecke-type double-sums in terms of Appell--Lerch functions. We give examples where the heuristic can be used as a guide to evaluate analogous triple-sums in terms of Appell--Lerch functions or false theta functions.Comment: 26 pages. Submitted to Hardy Ramanujan Journal for special volume in honour of Ramanuja

Splitting Appell functions in terms of single quotients of theta functions

Ramanujan's last letter to Hardy introduced the world to mock theta functions, and the mock theta function identities found in Ramanujan's lost notebook added to their intriguing nature. For example, we find the four tenth-order mock theta functions and their six identities. The six identities themselves are of a spectacular nature and were first proved by Choi. We also find over eight sixth-order mock theta functions in the lost notebook, but among their many identities there is only one relationship like those of the tenth-orders. Recently, three new identities for the sixth-order mock theta functions that are in the spirit of the six tenth-order identities were discovered. Here we present several families of tenth-order like identities for Appell functions, which are the building blocks of Ramanujan's mock theta functions.Comment: 31 page

Expressing qq-series in terms of building blocks of Hecke-type double-sums

We express recent double-sums studied by Wang, Yee, and Liu in terms of two types of Hecke-type double-sum building blocks. When possible we determine the (mock) modularity. We also express a recent $q$-hypergeometric function of Andrews as a mixed mock modular form

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