The B-B-G Transfer Principle for signature four
We show how the elliptic function ${\rm dn}_2$ of Shen leads to the signature four transfer principle of Berndt, Bhargava and Garvan.
math.CA↗
arXiv subjects
P. L. Robinson
Publications and source records attributed to P. L. Robinson.
Gauss $q$-ed from Heine cubed
We consider $q$-analytic derivations of the $q$-Gauss summation formula for a $\, _2ϕ_1$ that respect the symmetry in its upper parameters.
math.CA↗
The Berndt-Bhargava-Garvan Transfer Principle for signature three
We offer a new proof for the Berndt-Bhargava-Garvan Transfer Principle that connects the signature-three elliptic theory of Ramanujan to the classical elliptic theory.
math.CV↗
Modular transformations and the elliptic functions of Shen
We employ Weierstrassian modular transformations to compute fundamental periods for the elliptic functions ${\rm dn}_2$ and ${\rm dn}_3$ of Shen.
math.NT↗
A root elliptic function in signature four
We present a new approach to elliptic functions in signature four, offering a fresh perspective on work of Li-Chien Shen.
math.CA↗
Elliptic functions in signature four
We explore the relationships between two elliptic functions constructed by Shen in the signature four Ramanujan theory.
math.CV↗
On Weierstrass $\wp$ in signature 3
In his work on the Ramanujan theory of elliptic functions to alternative bases, Shen constructed two different elliptic functions in signature three; we determine the precise relationship between them, by precisely relating their coperiodic Weierstrass $\wp$ functions.
math.CV↗
Hypergeometric identities in elliptic signature six
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 nevertheless engender the corresponding hypergeometric identities of Ramanujan.
math.CV↗
The elliptic function ${\rm dn}_2$ of Shen
We analyze the elliptic function ${\rm dn}_2$ introduced by Li-Chien Shen, contributing to the Ramanujan theory of elliptic functions in signature four.
math.CV↗
The elliptic function ${\rm dn}_3$ of Shen
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.
math.CV↗
Elliptic functions from hypergeometric integrals
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 four. We determine precisely the signatures in which the Jacobian analogues or their squares are indeed elliptic.
math.CA↗
Nonelliptic functions from $F(\frac{1}{6}, \frac{5}{6} ; \frac{1}{2} ; \bullet)$
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 $F(\tfrac{1}{4}, \tfrac{3}{4}; \tfrac{1}{2} ; \bullet)$. We apply his methods to the hypergeometric function $F(\tfrac{1}{6}, \tfrac{5}{6}; \tfrac{1}{2} ; \bullet)$.
math.CV↗
Closed graphs and open maps
We offer a new perspective on the closed graph theorem and the open mapping theorem for separated barrelled spaces and fully complete spaces.
math.FA↗
The closed graph theorem is the open mapping theorem
We extend the closed graph theorem and the open mapping theorem to a context in which a natural duality interchanges their extensions.
math.FA↗
Elliptic functions and flotation
A paraboloid or a cone of density $ρ$ with vertical axis is released from rest into a liquid of density $ρ_0$. We determine the critical value of the ratio $ρ_0/ρ$ for subsequent full submersion.
math.GM↗
Elliptic functions from $F(\frac{1}{4}, \frac{3}{4} ; \frac{1}{2} ; \bullet)$
We reconsider the elliptic functions that are generated from the hypergeometric function $F(\tfrac{1}{4}, \tfrac{3}{4}; \tfrac{1}{2} ; \bullet)$ by Li-Chien Shen, presenting fresh proofs that do not require the use of theta functions.
math.CV↗
Elliptic functions from $F(\frac{1}{3}, \frac{2}{3} ; \frac{1}{2} ; \bullet)$
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.
math.CV↗
Higher Trigonometry: A Class Of Nonlinear Systems
We study the initial value problem '$s\,' = c^{p - 1}, \; c\,' = -s^{p - 1}; \; \; s(0) = 0, \; c(0) = 1$' (both as a real system and as a complex system) for each integer $p > 2$, considering separately the cases '$p$ even' and '$p$ odd'.
math.CA↗