Related papers: On Weierstrass $\wp$ in signature 3
We explore the relationships between two elliptic functions constructed by Shen in the signature four Ramanujan theory.
We analyze the elliptic function ${\rm dn}_3$ introduced by Li-Chien Shen, contributing to the Ramanujan theory of elliptic functions in signature three. A famous hypergeometric identity emerges from our analysis.
As a contribution to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has shown how analogues of the Jacobian elliptic functions may be derived from incomplete hypergeometric integrals in signatures three and…
Within the Ramanujan theories of elliptic functions, Li-Chien Shen constructed natural elliptic functions in signature three and signature four. When applied in signature six, the same constructions produce non-elliptic functions that…
We analyze the elliptic function ${\rm dn}_2$ introduced by Li-Chien Shen, contributing to the Ramanujan theory of elliptic functions in signature four.
We employ Weierstrassian modular transformations to compute fundamental periods for the elliptic functions ${\rm dn}_2$ and ${\rm dn}_3$ of Shen.
We present some properties of the Weierstrass $\wp$-function associated to the hexagonal (or triangular) lattice. In particular, with the help of an old theorem of I.N. Baker \cite{B} on the characterization of meromorphic solutions of the…
We present a new approach to elliptic functions in signature four, offering a fresh perspective on work of Li-Chien Shen.
In this paper, we introduce and study multiple $\wp$-functions, which generalize the classical Weierstrass $\wp$-function to iterated sums over lattice points, and we establish explicit formulas expressing them in terms of single…
We consider the generalized dual transformation for elliptic/hyperelliptic $\wp$ functions up to genus three. For the genus one case, from the algebraic addition formula, we deduce that the Weierstrass $\wp$ function has the SO(2,1) $\cong$…
The Weierstrassian $\wp, \zeta$ and $\sigma $ functions are generalized to ${\bf R}^{n}$. The $n=3$ and $n=4$ cases have already been used in gravitational and Yang-Mills instanton solutions which may be interpreted as explicit realizations…
In this paper, we present a general method for obtaining addition theorems of the Weierstrass elliptic function $\wp(z)$ in terms of given parameters. We obtain the classical addition theorem for the Weierstrass elliptic function as a…
As contributions to the Ramanujan theory of elliptic functions to alternative bases, Li-Chien Shen has developed families of elliptic functions from the hypergeometric functions $F(\tfrac{1}{3}, \tfrac{2}{3}; \tfrac{1}{2} ; \bullet)$ and…
In the theory of elliptic functions and elliptic curves, the Weierstrass $zeta$ function (which is essentially an antiderivative of the Weierstrass $\wp$ function) plays a prominent role. Although it is not an elliptic function, Eisenstein…
We present new addition formulae for the Weierstrass functions associated with a general elliptic curve. We prove the structure of the formulae in n-variables and give the explicit addition formulae for the 2- and 3-variable cases. These…
Li-Chien Shen developed a family of elliptic functions from the hypergeometric function $_2F_1(\frac{1}{3}, \frac{2}{3} ; \frac{1}{2} ; \bullet)$. We comment on this development, offering some new proofs.
A consistent notation for the Weierstrass elliptic function $\wp(z;g_{2},g_{3})$, for $g_{2} > 0$ and arbitrary values of $g_{3}$ and $\Delta \equiv g_{2}^{3} - 27 g_{3}^{2}$, is introduced based on the parametric solution for the motion of…
A class of exact solutions of the Skyrme model are obtained. They are described by the Weierstrass $\wp$-function or the Jacobi elliptic function. They are not solitonic but of wave character. They supply us with examples of the…
We investigate the first-order system `$s\,' = c^3, \, c\,' = - s^3; \, s(0) = 0, \, c(0) = 1$'. Its solutions have the property that $s \, c$, $s^2$ and $c^2$ extend to simply-poled elliptic functions, which we explicitly identify in terms…
We show how the elliptic function ${\rm dn}_2$ of Shen leads to the signature four transfer principle of Berndt, Bhargava and Garvan.