Related papers: On functions of Jacobi and Weierstrass (I)
In this note, we describe a general procedure to prove functional equations involving quasi-periodic functions. We give novel proofs for fundamental identities of Weierstrass sigma and Jacobi theta functions. Our method is based on the…
The paper is an essentially extended version of the work math.CA/0601371, supplemented with an application. We present new results in the theory of classical $\theta$-functions of Jacobi and $\sigma$-functions of Weierstrass: ordinary…
In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M.…
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.
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
This is an extended (factor 2.5) version of arXiv:math/0601371 and arXiv:0808.3486. We present new results in the theory of the classical $\theta$-functions of Jacobi: series expansions and defining ordinary differential equations (\odes).…
We evaluate some finite and infinite sums involving $q$-trigonometric and $q$-digamma functions. Upon letting $q$ approach $1$, one obtains corresponding sums for the classical trigonometric and the digamma functions. Our key argument is a…
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 propose a system of equations that defines Weierstrass--Jacobi's eta- and theta-constant series in a differentially closed way. This system is shown to have a direct relationship to a little-known dynamical system obtained by Jacobi. The…
We construct harmonic weak Maass forms that map to cusp forms of weight $k\geq 2$ with rational coefficients under the $\xi$-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished…
Jacobi theta functions with rational characteristics can be viewed as vector-valued Jacobi forms. Theta relations usually correspond to different constructions of certain Jacobi forms. From this observation, we extract a new approach, which…
Jacobi's theta relations among quartic products of theta functions are generalized to those of arbitrary $n$ products. Igusa's procedure of derivation is extended to prove such general theta relations, from which we obtain general addition…
We consider a generalization of Jacobi theta series and show that every such function is a quasi-Jacobi form. Under certain conditions we establish transformation laws for these functions with respect to the Jacobi group and prove such…
In this paper, we expand the theory of Weierstrassian elliptic functions by introducing auxiliary zeta functions $\zeta_\lambda$, zeta differences of first kind $\Delta_\lambda$ and second kind $\Delta_{\lambda,\mu}$ where…
The Jacobi and Weierstrass elliptic functions used to be part of the standard mathematical arsenal of physics students. They appear as solutions of many important problems in classical mechanics: the motion of a planar pendulum (Jacobi),…
Using the duplication formulas of the elliptic trigonometric functions of Gosper, we deduce some new special values for the first two Jacobi theta functions. At the end of the paper, we show how is it possible to extend our arguments and…
Is is shown in this paper that there is a connection between the Riemann zeta-function $\zf$ and the classical Jacobi's polynomials, i.e. the Legendre polynomials, Chebyshev polynomials of the first and the second kind,...
We collect some classical results related to analysis on the Riemann surfaces. The notes may serve as an introduction to the field: we suppose that the reader is familiar only with the basic facts from topology and complex analysis. the…
In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by…
Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we…