Related papers: On the Complete Evaluation of Jacobi Theta Functio…
By starting with Durand's double integral representation for a product of two Jacobi functions of the second kind, we derive an integral representation for a product of two Jacobi functions of the second kind in kernel form. We also derive…
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…
The Jacobi evolution method has been widely used in the QCD analysis of structure function data. However a recent paper claims that there are serious problems with its convergence and stability. Here we briefly review the evidence for the…
We present a new package Theta.jl for computing with the Riemann theta function. It is implemented in Julia and offers accurate numerical evaluation of theta functions with characteristics and their derivatives of arbitrary order. Our…
We calculate the Jacobian matrix of the dihedral angles of a generalized hyperbolic tetrahedron as functions of edge lengths and find the complete set of symmetries of this matrix.
A new method for continuing the usual Dirichlet series that defines the Riemann zeta function ${\zeta}(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.
The definitions and main properties of the Ihara and Bartholdi zeta functions for infinite graphs are reviewed. The general question of the validity of a functional equation is discussed, and various possible solutions are proposed.
In the article we outline the set of Matlab functions that enable the computation of elliptic Integrals and Jacobian elliptic functions for real arguments. Correctness, robustness, efficiency and accuracy of the functions are discussed in…
Dan Romik recently considered the Taylor coefficients of the Jacobi theta function around the complex multiplication point $i$. He then conjectured that the Taylor coefficients $d(n)$ either vanish or are periodic modulo any prime ${p}$;…
We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals $K$ and $E$. Our methods exploit the rich structures connecting complete elliptic integrals,…
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…
System of alternatively orthogonalized rational functions of Jacobi type on the half line $[1, \infty)$ is defined and its properties are established. Three subsystems of proper and mixed systems of rational functions with nice properties…
In this paper we introduce new generalizations of the zeta function, the Tricomi functions; their main properties are studied. This opens the way to a deeper, better application of these functions both in the theory of special functions,…
We give an elementary proof of the development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions.
In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta…
We consider the partial theta function $\theta (q,x):=\sum_{j=0}^{\infty}q^{j(j+1)/2}x^j$, where $x\in \mathbb{C}$ is a variable and $q\in \mathbb{C}$, $0<|q|<1$, is a parameter. We show that, for any fixed $q$, if $\zeta$ is a multiple…
Jacobi elliptic functions are flexible functions that appear in a variety of problems in physics and engineering. We introduce and describe important features of these functions and present a physical example from classical mechanics where…
We introduce a new set of algorithms to compute Jacobi matrices associated with measures generated by infinite systems of iterated functions. We demonstrate their relevance in the study of theoretical problems, such as the continuity of…
Despite the fact that there is a huge amount on papers and books devoted to the theory of Jacobian elliptic functions, very little is known when the modulus $k$ of these functions lies outside the unit interval $[0,1]$. In this note, we…
In this article we study properties of Ramanujan's mock theta functions that can be expressed in Lerch sums. We mainly show that each Lerch sum is actually the integral of a Jacobian theta function (here we show that for $\vartheta_3(t,q)$…