Questions:
- What are hypergeometrical series/functions?
- How are they used in Ramanujan's works?
- What are the applicants of them?
Apparently so, hypergeometrical functions appear in Ramanujan's works, and I would like to know what they are.
This one book did give this equation:$$1+\dfrac {\alpha\cdot\beta}{1\cdot\gamma}x+\dfrac {\alpha\beta(\alpha+1)(\beta+1)}{1\cdot2\cdot\gamma(\gamma+1)}x^2+\dfrac {\alpha\beta(\alpha+1)(\alpha+2)(\beta+1)(\beta+2)}{1\cdot2\cdot3\cdot\gamma(\gamma+1)(\gamma+2)}x^3+\cdots\&\text c$$ Which is convergent if $x<1$ and divegent if $x>1$. If $x=1$, then convergent if $\gamma-\alpha-\beta$ is positive, divergent if $\gamma-\alpha-\beta$ is negative, and divergent if $\gamma-\alpha-\beta$ is zero.
However, I still don't understand how to use them and how they are useful in continued fractions and $\pi$ formulas.
If we have $x=1$ and $\alpha=1,\beta=2,\gamma=5$, then$$F(\alpha,\beta,\gamma)=F(1,2,5)=1+\dfrac 25+\dfrac {12}{60}+\dfrac {144}{1260}+\dfrac {2880}{40320}+\cdots\&\text c$$
Which I don't see it accomplishing anything.