Gauss's Hypergeometric Theorem/Historical Note

Historical Note on Gauss's Hypergeometric Theorem
The proof shown above is a more detailed version of a proof by Srinivasa Ramanujan.

Based on Ramanujan's Notebook, as transcribed in Chapter $10$ of Berndt's book, Ramanujan's proof goes as follows:


 * Assume that $n$ and $x$ are integers with $n \ge 0$ and $n + x \ge 0$.
 * Expanding $\paren {1 + u}^{y + n}$ and $\paren {1 + \dfrac 1 u}^x$ in their formal binomial series and taking their product, we find that, if $a_n$ is the coefficient of $u^n$,


 * On the other hand, expanding $\paren {1 + u}^{x + y + n}$ in its binomial series and dividing by $u^x$, we find that


 * Comparing $\paren {8.2}$ and $\paren {8.3}$, we deduce $\paren {8.1}$.

As you can see, Ramanujan long jumped over several intermediate steps in $\paren {8.2}$, but his assertions were all correct.