With those mid-sem bells chiming, it is time for another update.
The following checkpoints have been reached:

1. The implementation of ellipsoidal harmonic function (also known as Lame's function): The first kind.
The following is the link to the pull request:
https://github.com/scipy/scipy/pull/3753
The implementation is in Cython and calls LAPACK subroutine. It is based on the python implementation by Knepley and Bardhan given here
Further the absence of Lame's function implementation by any many other libraries there is a challenge in preparation of an extensive test-suite. At present the output of the function for certain range of inputs is tested. The immediate next plan is to try improving the test-suite.
This will be followed by the implementation of Ellipsoidal harmonic function: The second kind.


2. Before this the spherical harmonic functions were improved by reimplementing them in Cython rather than python thus improving the speed.
The details having been elaborately touched in the previous post, are omitted here, saving people from the boredom due to redundancy. The pull request can be accessed from https://github.com/scipy/scipy/pull/3692

Thanks to the constant support of my brilliant mentors Pauli, Ralf and Stefan, (and I suppose, few miracles!) the progress is as per schedule. Hoping that this pace will be maintained or even better, quickened; post mid-sem evaluation!

Signing off till next time,
Janani

2 weeks over! That was fast..
And I have sent my pull request finishing the first of 4 sub-parts of my project.
The link to the pull request is https://github.com/scipy/scipy/pull/3692.
Here the function to implement spherical harmonics has been rewritten in Cython. It uses lpmv fortran sub-routine (with the help of the existing wrapper function pmv_wrap) to calculate associated Legendre Function.
Initially the function to calculate spherical harmonics was written in python which called lpmn function to calculate the associated Legendre function. Lpmn function in turn uses lpmn fortran routine to calculate the function. The routine iteratively evaluates associated Legendre function for all orders and degrees m and n. While the routine lpmv avoids the iteration by using recursion.
The usage of a different routine and rewriting the code in cython has given a huge leap in the speed of the function which is shown in a simple benchmark performed by Pauli,
Before:
 In [4]: %timeit sph_harm(50, 50, 0.5, 0.9)
10000 loops, best of 3:
81.3 µs per loop
 In [5]: x = np.linspace(0, 1, 2000)
 In [6]: %timeit sph_harm(50, 50, 0.5, x)
 10 loops, best of 3:
127 ms per loop

After:
 In [2]: %timeit sph_harm(50, 50, 0.5, 0.9)
100000 loops, best of 3:
 4.52 µs per loop
 In [3]: x = np.linspace(0, 1, 2000)
In [4]: %timeit sph_harm(50, 50, 0.5, x)
1000 loops, best of 3: 1.87 ms per loop

Now the next part of the project is to implement the ellipsoidal harmonic functions. It is kind of challenging as they are complex and they havent been implemented much. My next 2-3 weeks are dedicated to the implementation of the both kinds of ellipsoidal functions, thus meeting my mid sem deadline.