Jacobi Theta Research Articles

Convexity of quotients of theta functions

For fixed u and v such that 0 ⩽ u < v < 1 / 2 , the monotonicity of the quotients of Jacobi theta functions, namely, θ j ( u | i π t ) / θ j ( v | i π t ) , j = 1 , 2 , 3 , 4 , on 0 < t < ∞ has been established in the previous works of A.Yu. Solynin, K. Schiefermayr, and Solynin and the first author. In the present paper, we show that the quotients θ 2 ( u | i π t ) / θ 2 ( v | i π t ) and θ 3 ( u | i π t ) / θ 3 ( v | i π t ) are convex on 0 < t < ∞ .

On the modular behaviour of the infinite product [formula omitted

Let q = e 2 π i τ , ℑ τ > 0 , x = e 2 π i ξ ∈ C and ( x ; q ) ∞ = ∏ n ⩾ 0 ( 1 − x q n ) . Let ( q , x ) ↦ ( q ⁎ , ι q x ) be the classical modular substitution given by q ⁎ = e − 2 π i / τ and ι q x = e 2 π i ξ / τ . The main goal of this Note is to study the “modular behaviour” of the infinite product ( x ; q ) ∞ , this means, to compare the function defined by ( x ; q ) ∞ with that given by ( ι q x ; q ⁎ ) ∞ . Inspired by the work [16] of Stieltjes (1886) on some semi-convergent series, we are led to a “closed” analytic formula for the ratio ( x ; q ) ∞ / ( ι q x ; q ⁎ ) ∞ by means of the dilogarithm combined with a Laplace type integral, which admits a divergent series as Taylor expansion at log q = 0 . Thus, we can obtain an expression linking ( x ; q ) ∞ to its modular transform ( ι q x ; q ⁎ ) ∞ and which contains, in essence, the modular formulae known for Dedekindʼs eta function, Jacobi theta function and also for certain Lambert series. Among other applications, one can remark that our results allow one to interpret Ramanujanʼs formula (Berndt, 1994) [5, Entry 6, p. 265 & Entry 6′, p. 268] (see also Ramanujan, 1957 [10, pp. 365 & 284]) as being a convergent expression for the infinite product ( x ; q ) ∞ .

ZERO-FREE REGIONS OF A q-ANALOGUE OF THE COMPLETE RIEMANN ZETA FUNCTION

A q-analogue of the complete Riemann zeta function presented in this paper is defined by the q-Mellin transform of the Jacobi theta function. We study zero-free regions of the q-zeta function. As a by-product, we show that the Riemann zeta function does not vanish in a sub-region of the critical strip.

Une généralisation de la formule du triple produit de Jacobi et quelques applications

If A n ≠ 0 for all n ∈ Z , we show the series with 2 variables Q ( x , y ) = ∑ n ∈ Z A n x n y n ( n + 1 ) / 2 factorizes formally in an infinite triple product, which generalizes the Jacobiʼs formula. Let ρ o be the positive root of ∑ k = 1 ∞ ρ k 2 = 1 / 2 , we prove the convergence of the factorization of Q for x ∈ C ⁎ and | y | < ρ o 2 Ω − 1 with Ω = sup n ∈ Z | A n − 1 A n + 1 / A n 2 | . We deduce that if Ω < ρ o 2 = 0.2078 … each zero of the Laurent series f ( x ) = ∑ n ∈ Z A n x n can be explicitly calculated as the sum or the inverse of the sum of series, whose terms are polynomial expressions of A n − 1 A n + 1 / A n 2 . If the previous inequality is wide and f ( x ) real, then all its zeros are real numbers. An other application is when you know the triple product factorization of Q ( x , y ) by another way than described in the note, to identify them. So with the Jacobi theta function, we obtained a new identity for the sum of divisors σ ( n ) of an integer.

A Maass Lifting of $\Theta^3$ and Class Numbers of Real and imaginary Quadratic Fields

We give an explicit construct of a harmonic weak Maass form $F_{\Theta}$ that is a lift of $\Theta^3$, where $\Theta$ is the classical Jacobi theta function. Just as the Fourier coefficients of $\Theta^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the holomorphic part of $F_{\Theta}$ are associated to class numbers of real quadratic fields.

Casimir energy and brane stability

We investigate the role of Casimir energy as a mechanism for brane stability in five-dimensional models with the fifth dimension compactified on an S 1 / Z 2 orbifold, which includes the Randall–Sundrum two brane model (RS1). We employ a ζ -function regularization technique utilizing the Schwinger proper time method and the Jacobi theta function identity to calculate the one-loop effective potential. We show that the combination of the Casimir energies of a scalar Higgs field, the three generations of Standard Model fermions and one additional massive non-SM scalar in the bulk produces a non-trivial minimum of the potential. In particular, we consider a scalar field with a coupling in the bulk to a Lorentz violating vector particle localized to the compactified dimension. Such a scalar may provide a natural means of fine tuning needed for stabilization of the brane separation. Finally, we briefly review the possibility that Casimir energy plays a role in generating the currently observed epoch of cosmological inflation by examining a simple five-dimensional anisotropic metric.

Modeling of a voltammetric experiment in a limiting diffusion space

A voltammetric experiment confined in a limiting diffusion space is analyzed theoretically governed by conventional or time-anomalous factional diffusion under conditions of cyclic and square-wave voltammetry. The solution for conventional diffusion is derived by means of the Jacobi theta function $$ \Theta \left( {{a^2}/{\pi^2}t} \right)\left( {a = L{D^{ - 1/2}}} \right. $$ , where L is the thickness of the finite diffusion space, D is the diffusion coefficient, and t is the time of the experiment) and compared with the solution frequently used in the literature expressed in the form Θ(a −2 t). For L → ∞, the present solution converges to the one for the semi-infinite diffusion, thus being of a general applicability for both finite and semi-infinite diffusion. Hence, the mathematical model for simulation of both cyclic and square-wave voltammetric experiment provides significant advances in terms of simulation time and accuracy compared to the previous model based on the modified step-function method Mirceski (J Phys Chem B 108:13719, 2004). For the fractional diffusion experiment, the solution is derived by combining an infinite series and the Wright function $$ \phi \left( { - \alpha /2,\alpha /2; - 2a{\xi^{ - 1/2}}{t^{ - \alpha /2}}} \right) $$ , where α is the time fractional parameter ranging over the interval $$ 0 < \alpha < {1} $$ , and ξ = 1 s1−α is the auxiliary constant. The voltammetric properties of the experiment controlled by fractional diffusion are comparable for both finite and semi-infinite diffusion.

Le module de continuité des valeurs au bord des fonctions propres et des formes automorphes

Let Γ be a Fuchsian group acting on the hyperbolic plane ℍ. Any eigenfunction of the Laplacian, any automorphic form on an hyperbolic surface ℍ/ Γ induce a distribution on the boundary ∂ ℍ. This distribution is the derivative of a certain order of a fonction F on ∂ ℍ : the derivative of order 1 in the case of bounded eigenfunctions, the derivative of order in the case of cuspidal automorphic forms of weight k . For cuspidal eigenfunctions (resp. for cuspidal automorphic forms) the optimal Hölder exponent of F at a point ξ ∈ ∂ ℍ can be computed exactly when ℍ/ Γ has finite volume : this exponent depends only on the eigenvalue of the function (resp. on the weight of the automorphic form) and on the fact that the ray oξ be recurrent or not ; in particular F is not differentiable at any point, except possibly at the cusps, depending on the eigenvalue (resp. on the weight). For regular but non-cuspidal automorphic forms, there is a continuous family of possibilities for the modulus of continuity. We study in details the case of automorphic forms of weight (like the Jacobi theta function, whose associate function F on ∂ ℍ has imaginary part the Riemann function : using properties of the geodesic flow such as the Khintchine-Sullivan theorem, we show that at almost all ξ a modulus of continuity is , for any є &gt; 0.

A variant of Jacobi type formula for Picard curves

The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves E ( λ ) : w 2 = z ( z - 1 ) ( z - λ ) . In other words, this formula says that the modular form ϑ 00 4 ( τ ) with respect to the principal congruence subgroup Γ ( 2 ) of PSL ( 2 , Z ) has an expression by the Gauss hypergeometric function F ( 1 / 2 , 1 / 2 , 1 ; 1 - λ ) via the inverse of the period map for the family of elliptic curves E ( λ ) (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves C ( λ 1 , λ 2 ) : w 3 = z ( z - 1 ) ( z - λ 1 ) ( z - λ 2 ) , those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for C ( λ 1 , λ 2 ) is established in [S] and our modular form ϑ 0 3 ( u , v ) (for the definition, see (2.7)) is defined on a two dimensional complex ball D = { 2 Re v + | u | 2 < 0 } , that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function F 1 ( 1 / 3 , 1 / 3 , 1 / 3 , 1 ; 1 - λ 1 , 1 - λ 2 ) .

Exact Solution of the Six‐Vertex Model with Domain Wall Boundary Conditions: Antiferroelectric Phase

AbstractWe obtain the large‐n asymptotics of the partition function Zn of the six‐vertex model with domain wall boundary conditions in the antiferroelectric phase region, with the weights a = sinh(γ − t), b = sinh(γ + t), c = sinh(2γ), |t| &lt; γ. We prove the conjecture of Zinn‐Justin, that as n → ∞, Zn = Cϑ4(nω)F [1 + O(n−1)], where ω and F are given by explicit expressions in γ and t, and ϑ4(z) is the Jacobi theta function. The proof is based on the Riemann‐Hilbert approach to the large‐n asymptotic expansion of the underlying discrete orthogonal polynomials and on the Deift‐Zhou nonlinear steepest‐descent method. © 2009 Wiley Periodicals, Inc.

A τ-function solution of the sixth painlevé transcendent

We represent and analyze the general solution of the sixth Painleve transcendent \( \mathcal{P}_6 \) in the Picard-Hitchin-Okamoto class in the Painleve form as the logarithmic derivative of the ratio of τ-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the \( \mathcal{P}_6 \) equation and the uniformization of algebraic curves and present examples.

On exact simulation algorithms for some distributions related to Jacobi theta functions

We develop exact random variate generators for several distributions related to the Jacobi theta function. These include the distributions of the maximum of a Brownian bridge, a Brownian meander and a Brownian excursion, and distributions of certain first passage times of Bessel processes. The algorithms are based on the alternating series method. Furthermore, we survey various distributional identities and point out ways of dealing with generalizations of these basic distributions.

A class of exact solutions of the generalized nonlinear Schrödinger equation

The generalized nonlinear Schrodinger equation, studied in this paper, is an important model of nonlinear excitations of matter waves in Bose—Einstein condensates. Through a natural transformation and the Hirota bilinear method, N bright and dark soliton solutions are obtained. Based on Jacobi theta functions, periodic-like solutions, which are periodic in x, are also presented.

A resummation formula for collapse and revival in the Jaynes–Cummings model

We present a new approach to the study of the function of atomic inversion in the model of interaction of a single two-level atom with a single mode of the quantized electromagnetic field in the coherent state in an ideal resonator. The approach suggested is based on an application of certain number-theoretic results to the approximation of the trigonometric sums of a special form, in particular of the functional equation of the Jacobi theta function. New asymptotic formulae for the atomic inversion are found. The asymptotics that we obtain make it possible to predetermine the details of the behavior of the inversion on various time intervals depending on the parameters of the system.

The Rogers–Ramanujan continued fraction and a new Eisenstein series identity

With two elementary trigonometric sums and the Jacobi theta function θ 1 , we provide a new proof of two Ramanujan's identities for the Rogers–Ramanujan continued fraction in his lost notebook. We further derive a new Eisenstein series identity associated with the Rogers–Ramanujan continued fraction.

Addition formulas for Jacobi theta functions, Dedekind's eta function, and Ramanujan's congruences

Previously, we proved an addition formula for the Jacobi theta function, which allows us to recover many important classical theta function identities. Here, we use this addition formula to derive a curious theta function identity, which includes Jacobi�s quartic identity and some other important theta function identities as special cases. We give new series expansions for ?2(t), ?6(t), ?8(t), and ?10(t), where ?(t) is Dedekind�s eta function. The series expansions for ?6(t) and ?10(t) lead to simple proofs of Ramanujan�s congruences p(7n + 5) = 0 (mod 7) and p(11n + 6) = 0 (mod 11), respectively.

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type I. The Eigenfunction Identities

In this series of papers we study Hilbert-Schmidt integral operators acting on the Hilbert spaces associated with elliptic Calogero-Moser type Hamiltonians. As shown in this first part, the integral kernels are joint eigenfunctions of differences of the latter Hamiltonians. On the relativistic (difference operator) level the kernel is built from the elliptic gamma function, whereas the building block in the nonrelativistic (differential operator) limit is basically the Weierstrass sigma-function. For the AN−1 case we consider all of the commuting Hamiltonians at once, the eigenfunction properties reducing to a sequence of elliptic identities. For the BCN case we only treat the defining Hamiltonians. The functional identities encoding the eigenfunction properties have a remarkable corollary in the relativistic BC1 case: They imply that the sum over eight-fold products of the four Jacobi theta functions is invariant under the Weyl group of E8.

Applications of an advanced differential equation in the study of wavelets

A new wavelet family K ( t ) is discussed which represents a natural range of continuous pulse waveforms, deriving from the theory of multiplicatively advanced/delayed differential equations. K satisfies: all moments of K vanish; the Fourier transform of K relates to the Jacobi theta function; and K generates a wavelet frame for L 2 ( R ) . Estimates on the frame bounds as well as the translation parameter are provided.

Identities of arithmetic type between values of the theta function associated to a lattice in ℝ d and its derivatives

In this paper, we introduce a notion of similarly self dual lattice in a d-dimensional Euclidean space and a classical Jacobi theta function is associated to such a lattice. We establish identities of arithmetic type between values of this theta function and its successive derivatives. This work can be related to the spectral theory of the Landau operators.

GENERALIZED mth ORDER JACOBI THETA FUNCTIONS AND THE MACDONALD IDENTITIES

We describe an mth order generalization of Jacobi's theta functions and use these functions to construct classes of theta function identities in multiple variables. These identities are equivalent to the Macdonald identities for the seven infinite families of irreducible affine root systems. They are also equivalent to some elliptic determinant evaluations proven recently by Rosengren and Schlosser.