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Andrews-Dyson-Hickerson, Cohen build a striking relation between q-hypergeometric series, real quadratic fields, and Maass forms. Thanks to the works of Lewis-Zagier and Zwegers we have a complete understanding on the part of these…
We describe the construction of vector valued modular forms transforming under a given congruence representation of the modular group SL(2,Z) in terms of theta series. We apply this general setup to obtain closed and easily computable…
We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion at any point of the theta divisor. We…
We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The \textit{level-$\ell$ Appell functions} $K_\ell$ satisfy open quasiperiodicity relations…
We derive lattice invariants from the heat flux of a lattice. Using systems of harmonic polynomials, we obtain sums of products of spherical theta functions which give new invariants of integer lattices which are modular forms. In…
In this paper, we establish Kronecker limit type formulas for the generalized Mordell--Tornheim zeta function $\Theta(r,s,t,x)$ as a function of the third variable, in terms of Riemann-zeta and Gamma values. We also give series evaluations…
We give a new theta-function identity, a special case of which is utilised to prove Kawanaka's Macdonald polynomial conjecture. The theta-function identity further yields a transformation formula for multivariable elliptic hypergeometric…
Making use of inverse Mellin transform techniques for analytical continuation, an elegant proof and an extension of the zeta function regularization theorem is obtained. No series commutations are involved in the procedure; nevertheless the…
The topological zeta function of a matroid is a rational function as well as a valuative invariant of the matroid, encoding rich combinatorial information. We analyze topological zeta functions of matroids from the vantage point of several…
We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on…
We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of ($S$-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta…
In the theory of harmonic Maass forms and mock modular forms, mock theta functions are distinguished examples which arose from $q$-hypergeometric examples of Ramanujan. Recently, there has been a body of work on higher depth mock modular…
In this paper, we investigate new relationships for bilateral series related to two-parameter mock theta functions, which lead to many identities concerning the bilateral mock theta functions. In addition, interesting relations between the…
Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses the completion through the holomorphic…
This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be…
This manuscript introduces a general multisection identity expressed equivalently in terms of infinite double products and/or infinite double series, from which several new product or summation identities involving special functions…
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…
We extend a certain type of identities on sums of $I$-Bessel functions on lattices, previously given by G. Chinta, J. Jorgenson, A. Karlsson and M. Neuhauser. Moreover we prove that, with continuum limit, the transformation formulas of…
We consider matrices with entries in a local ring, Mat(m,n;R). Fix an action of group G on Mat(m,n;R), and a subset of allowed deformations, \Sigma in Mat(m,n;R). The standard question (along the lines of Singularity Theory) is the…
Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which not only gives, as a special case, a famous modular relation between the Rogers-Ramanujan functions $G(q)$ and $H(q)$ but also a relation between two fifth order mock…