NAME : Dr Heng Huat Chan

 ADDRESS: National University of Singapore
          Department of Mathematics 
          2 Science Drive 2,
          Singapore 117543, Republic of Singapore
 
 EMAIL ADDRESS: chanhh@math.nus.edu.sg

 TITLE OF TALK: A new class of series for $\dfrac{1}{\pi}$ 


 ABSTRACT OF TALK:
 Let $f(-q ) = \prod_{k=1}^\infty (1-q^k), |q|<1$. On page 212 of his 
 Lost Notebook, S. Ramanujan defined the function  
 $$
 \lambda_n 
 = \dfrac{e^{\pi/2\sqrt{n/3}}}{3\sqrt{3}}
   \dfrac{f^6(e^{-\pi\sqrt{n/3}})}{f^6(e^{-\pi\sqrt{3n}})}
 $$
 and recorded its values for $n=1, 9, 17, 25, 33, 41, 49, 73,$ 97, and 121. 
 The proofs of these evaluations have recently been given by B.C. Berndt, 
 S.Y. Kang, L.C. Zhang and the author. In this talk, I will show how the 
 known values of $\lambda_n$ can be used to establish new series for 
 $\tfrac{1}{\pi}$. An example of such a series, in the case when $n=9$, is 
 $$
 \dfrac{4}{\pi\sqrt{3}} 
    =\sum_{k=0}^\infty (5k+1)
        \dfrac{\left(\tfrac{1}{3}\right)_k\left(\tfrac{2}{3}\right)_k
    \left(\tfrac{1}{2}\right)_k}{(1)_k^3}\left(-\dfrac{9}{16}\right)^k,
  $$
  where $(a)_k = (a)(a+1)(a+2)\cdots (a+k-1).$