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Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and…
We prove the transformation laws of the four Jacobi theta functions using Gordon's proof for the transformation law of the Dedekind eta function.
The modular transformations of Ramanujan's tenth order mock theta functions are computed, beginning from Choi's Hecke-type identites and using Zwegers' results on indefinite theta series. Explicit completions and shadows are found as an…
In this paper, we derive a unified generalization of Ramanujan's transformation identities for the theta function $f(a,b)$, originally appearing in Ramanujan's Notebooks, Parts~III and IV. Using an approach based on residue-class…
The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's $_1\psi_1$ summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta…
In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular, we answer and improve some problems suggested and…
A new class of integrals involving the confluent hypergeometric function ${}_1F_{1}(a;c;z)$ and the Riemann $\Xi$-function is considered. It generalizes a class containing some integrals of S. Ramanujan, G.H. Hardy and W.L. Ferrar and gives…
Let $X$ be a compact Riemann surface of genus $g$. Jacobi's inversion theorem states that the Abel-Jacobi map $\varphi : X^{(g)} \longrightarrow J(X)$ is surjective, where $X^{(g)}$ is the symmetric product of $X$ of degree $g$ and $J(X)$…
Properties of the Jacobi Theta3-function and its derivatives under discrete Fourier transforms are investigated, and several interesting results are obtained. The role of modulo N equivalence classes in the theory of Theta-functions is…
In this Ph.D. thesis, written under the direction of D.B. Zagier and R.W. Bruggeman, we study the mock theta functions, that were introduced by Ramanujan. We show how they can be interpreted in the theory of (real-analytic) modular forms.…
We use the Jacobi theta function to give a representation of the modulus of the Riemann $\xi$ function. Based on this modulus representation, we show that the Riemann hypothesis is equivalent to the validity of a family of polynomial…
Eigenvectors of the discrete Fourier transform can be expressed using Ramanujan theta functions. New theta function identities, Ramanujan theta function identities, and generating functions for the quadratic numbers are a consequence.
In this short paper, we find the transformation formula for the theta series under the action of the Jacobi modular group on the Siegel-Jacobi space. This formula generalizes the formula (5.1) obtained by Mumford in his book[p.189, Tata…
In this paper, we generalize Rademacher's proof of the transformation law of the eta function to the Jacobi theta function using Residue calculus.
In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.
We consider integral and series transformations, which are associated with Ramanujan's identities, involving various arithmetic functions and a ratio of products of Riemann's zeta functions of different arguments. Reciprocal inversion…
In 'The Lost Notebook and Other Unpublished Papers' of Ramanujan are present some manuscripts of Ramanujan in the handwriting of G. N. Watson which are 'copied from loose papers'. We present a proof of a beautiful formula of Ramanujan in…
We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's…
Using numerical, theoretical and general methods, we construct evaluation formulas for the Jacobi $\theta$ functions. Some of our results are conjectures, but are verified numerically.
Unary theta functions have played a significant role in the theory of holomorphic modular forms and modular $L$-functions. A partial theta functions is defined analogously, but the sum is over part of the integer lattice. Such sums fail to…