A theta function identity of degree eight and Eisenstein series identities

In this paper we prove a theta function identity of degree eight using the theory of elliptic theta functions and the method of asymptotic analysis. This identity allows us to derive some curious Eisenstein series identities. We prove a new addition formula for theta functions which allows us to give an extension of the Hirschhorn septuple product identity.     

Classification of the space spanned by theta series and applications

We determine a class of functions spanned by theta series of higher degree. We give two applications: A simple proof of the inversion formula of such theta series and a classification of skew-holomorphic Jacobi forms.

     

Finite and Infinite Order Differential Relations for Theta Functions

We give different linear and nonlinear differential relations on Jacobi theta functions with more emphasis on the nonlinear differential equation of the third order of Jacobi. We present different points of view with a special attention to the role played by the second order linear differential equations, and their link to the Riccati equation and the Schwarzian equation. We also study an identity for theta functions resulting from the action of certain infinite order differential operators.     

Geometry of the finite fourier transform

We provide a geometric interpretation for the multidimensional finite Fourier transform and give a reformulation of the Macwilliams identitity in terms of theta functions.     

A three-term theta function identity and its applications

In this paper, we establish a three-term theta function identity using the complex variable theory of elliptic functions. This simple identity in form turns out to be quite useful and it is a common origin of many important theta function identities. From which the quintuple product identity and one general theta function identity related to the modular equations of the fifth order and many other interesting theta function identities are derived. We also give a new proof of the addition theorem for the Weierstrass elliptic function ℘. An identity involving the products of four theta functions is given and from which one theta function identity by McCullough and Shen is derived. The quintuple product identity is used to derive some Eisenstein series identities found in Ramanujan's lost notebook and our approach is different from that of Berndt and Yee. The proofs are self contained and elementary.     

Ramanujan’s 1 ψ 1 summation, Hecke-type double sums, and Appell–Lerch sums

We use a specialization of Ramanujan??s ₁ ?? ₁ summation to give a new proof of a recent formula of Hickerson and Mortenson which expands a special family of Hecke-type double sums in terms of Appell?CLerch sums and theta functions.     

An addition formula for the Jacobian theta function and its applications

In this paper, we prove an addition formula for the Jacobian theta function using the theory of elliptic functions. It turns out to be a fundamental identity in the theory of theta functions and elliptic function, and unifies many important results about theta functions and elliptic functions. From this identity we can derive the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities; and all of which are as striking as Winquist's identity. This identity allows us to give a new proof of the addition formula for the Weierstrass sigma function. A new identity about the Ramanujan cubic elliptic function is given. The proofs are self contained and elementary.     

New lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-Hamiltonian structure

We study restricted multiple three wave interaction system by the inverse scattering method. We develop the algebraic approach in terms of classical r-matrix and give an interpretation of the Poisson brackets as linear r-matrix algebra. The solutions are expressed in terms of polynomials of theta functions. In particular case for n = 1 in terms of Weierstrass functions.      

Opers and theta functions

We construct natural maps (the Klein and Wirtinger maps) from moduli spaces of semistable vector bundles over an algebraic curve X to affine spaces, as quotients of the nonabelian theta linear series. We prove a finiteness result for these maps over generalized Kummer varieties (moduli space of torus bundles), leading us to conjecture that the maps are finite in general. The conjecture provides canonical explicit coordinates on the moduli space. The finiteness results give low-dimensional parametrizations of Jacobians (in for generic curves), described by 2Θ functions or second logarithmic derivatives of theta.We interpret the Klein and Wirtinger maps in terms of opers on X. Opers are generalizations of projective structures, and can be considered as differential operators, kernel functions or special bundles with connection. The matrix opers (analogues of opers for matrix differential operators) combine the structures of flat vector bundle and projective connection, and map to opers via generalized Hitchin maps. For vector bundles off the theta divisor, the Szegö kernel gives a natural construction of matrix oper. The Wirtinger map from bundles off the theta divisor to the affine space of opers is then defined as the determinant of the Szegö kernel. This generalizes the Wirtinger projective connections associated to theta characteristics, and the associated Klein bidifferentials.     

Jacobi Theta Functions over Number Fields

We use Jacobi theta functions to construct examples of Jacobi forms over number fields. We determine the behavior under modular transformations by regarding certain coefficients of the Jacobi theta functions as specializations of symplectic theta functions. In addition, we show how sums of those Jacobi theta functions appear as a single coefficient of a symplectic theta function.     

Lambert series and Ramanujan

Lambert series are of frequent occurrence in Ramanujan's work on elliptic functions, theta functions and mock theta functions. In the present article an attempt has been made to give a critical and up-to-date account of the significant role played by Lambert series and its generalizations in further development and a better understanding of the works of Ramanujan in the above and allied areas.     

L-functions and theta correspondence for classical groups

Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the L-functions and local theta correspondence.     

On nested simple recursion

We give a novel proof that primitive recursive functions are closed under nested simple recursion. This new presentation is supplied with a detailed proof which can be easily formalized in small fragments of Peano Arithmetic.     

Theta Functions with Harmonic Coefficients over Number Fields

We investigate theta functions attached to quadratic forms over a number field K. We establish a functional equation by regarding the theta functions as specializations of symplectic theta functions. By applying a differential operator to the functional equation, we show how theta functions with harmonic coefficients over K behave under modular transformations.     

Automorphic forms and differentiability properties

We consider Fourier series given by a type of fractional integral of automorphic forms, and we study their local and global properties, especially differentiability and fractal dimension of the graph of their real and imaginary parts. In this way we can construct fractal objects and continuous non-differentiable functions associated with elliptic curves and theta functions.

     

Capacité de la réunion de trois intervalles et fonctions thêta de genre 2

We give a formula for the capacity of an union of three intervals. In the case of two intervals such a formula has been found by N.I. Achieser. The formula proposed here uses genus two theta functions and we recover Achieser's formula when we degenerate an interval or when there is a symmetry.     

A Diller-Nahm-style functional interpretation of $\hbox{\sf KP} \omega$

The Dialectica-style functional interpretation of Kripke-Platek set theory with infinity () given in [1] uses a choice functional (which is not a definable set function of (). By means of a Diller-Nahm-style interpretation (cf. [4]) it is possible to eliminate the choice functional and give an interpretation by set functionals primitive recursive in . This yields the following characterization: The class of -definable set functions of coincides with the collection of set functionals of type 1 primitive recursive in . Received: 26 August 1998     

A set of axioms for nonstandard extensions

We give an axiomatic characterization for complete elementary extensions, that is, elementary extensions of the first‐order structure consisting of all finitary relations and functions on the underlying set. Such axiom systems have been studied using various types of primitive notions (e.g., [1, 3, 6]). Our system uses the notion of partial functions as primitive. Properties of nonstandard extensions are derived from five axioms in a rather algebraic way, without the use of metamathematical notions such as formulas or satisfaction. For example, when applied to the real number system, it provides a complete framework for developing nonstandard analysis based on hyperreals without having to construct them and without any use of logic. This has possible pedagogical and expository applications as presented in, e.g., [5], [6]. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the asymptotics of some partial theta functions

Using the Euler–Maclaurin sum formula, we develop an asymptotic expansion for a fairly general sum of exponentials, which when specialized includes some common partial theta functions. Some conjectured asymptotic expansions for relevant integrals are given. We give a simple proof of a theorem by Bruce Berndt and Byungchan Kim generalizing a result found in Ramanujan’s second notebook.     

Matrices with maximum kth local exponent in the class of doubly symmetric primitive matrices

Let A be a primitive matrix of order n, and let k be an integer with 1?k?n. The kth local exponent of A, is the smallest power of A for which there are k rows with no zero entry. We have recently obtained the maximum value for the kth local exponent of doubly symmetric primitive matrices of order n with 1?k?n. In this paper, we use the graph theoretical method to give a complete characterization of those doubly symmetric primitive matrices whose kth local exponent actually attain the maximum value.