20 research outputs found
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
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 , 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 , 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
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 for levels .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
Expressing -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 -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
-series with mixed mock modular behaviour.Comment: To be published in Advances in Mathematic
Hecke-type double sums, Appell-Lerch sums, and mock theta functions (I)
By developing a connection between partial theta functions and Appell-Lerch
sums, we find and prove a formula which expresses Hecke-type double sums in
terms of Appell-Lerch sums and theta functions. Not only does our formula prove
classical Hecke-type double sum identities such as those found in work Kac and
Peterson on affine Lie Algebras and Hecke modular forms, but once we have the
Hecke-type forms for Ramanujan's mock theta functions our formula gives
straightforward proofs of many of the classical mock theta function identities.
In particular, we obtain a new proof of the mock theta conjectures. Our formula
also applies to positive-level string functions associated with admissable
representations of the affine Lie Algebra as introduced by Kac and
Wakimoto
The genomic landscape of balanced cytogenetic abnormalities associated with human congenital anomalies
Despite the clinical significance of balanced chromosomal abnormalities (BCAs), their characterization has largely been restricted to cytogenetic resolution. We explored the landscape of BCAs at nucleotide resolution in 273 subjects with a spectrum of congenital anomalies. Whole-genome sequencing revised 93% of karyotypes and demonstrated complexity that was cryptic to karyotyping in 21% of BCAs, highlighting the limitations of conventional cytogenetic approaches. At least 33.9% of BCAs resulted in gene disruption that likely contributed to the developmental phenotype, 5.2% were associated with pathogenic genomic imbalances, and 7.3% disrupted topologically associated domains (TADs) encompassing known syndromic loci. Remarkably, BCA breakpoints in eight subjects altered a single TAD encompassing MEF2C, a known driver of 5q14.3 microdeletion syndrome, resulting in decreased MEF2C expression. We propose that sequence-level resolution dramatically improves prediction of clinical outcomes for balanced rearrangements and provides insight into new pathogenic mechanisms, such as altered regulation due to changes in chromosome topology