Li-Chien Shen

On Ramanujan’s identities involving the Eisenstein series and hyper-geometric series


October 20, 2015, 3:00 — 3:50pm at LIT 368.


We shall study the properties of elliptic functions based on the differential equations
\(y’^2 = T_n(y) – (1 – 2\mu^2); \)
where \(\mu^2\) is a constant, \(T_n(x)\) are the Chebyshev polynomials with \(n = 3; 4; 6\) and the initial condition for each case is to be given later. The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function and can be used to construct theories of elliptic functions based on \({}_2F_1(1/4; 3.4; 1; z)\) and \({}_2F_1(1/3; 2/3; 1; z)\) and \({}_2F_1(1/6; 5/6; 1; z)\). From the perspective of differential equations, they are natural analogues of the classical elliptic functions of Jacobi and Weierstrass derived, respectively, from the solutions of the differential equations

\(y’2 = (1- y^2)(1 – k^2y^2);\;\; y(0) = 0\)

\(y’^2 = 4y^3 – g_3y – g_3; y(0) = \infty.\)