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Srinivasa Aiyangar Ramanujan

Some Special Srinivasa Aiyangar Ramanujan's Series and Formulas
(1887-1920)
 
 
 

  •     ,

 

where and for each positive integer k,
 
 
  •  

 

 
  • .

 

 
  •  .

 

 
  •  .

 

 
  • .

 

 

  • .

 

 
  • .

 

 
  • .

 

 
  • .

 

 
  • .

 

 
  • , for all , where is the Gamma Function.
 
 
  • Define , which is called Ramanujan Theta Function. Special values include

 

 
.
 
 
  • Hypergeometric Identity

 

.
 
 
  • Ramanujan’s Master Theorem

 

then
 
for small of x = 0.
 
 
 

 

 

References
Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, 1985.
Berndt, B. C. Ramanujan's Notebooks: Part II. New York: Springer-Verlag, 1989.
Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.
Berndt, B. C. and Bhargava, S. ``A Remarkable Identity Found in Ramanujan's Third Notebook.'' Glasgow Math. J, 1992.
D.V. Chudnovsky and G.V. Chudnovsky. Approxiamtions and Complex Multiplication According to Ramanujan. San Diego, CA: Academic Press, 1988.