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 $(8.2)$ and $(8.3)$, we deduce $(8.1)$.

As can be seen, Ramanujan jumped over several intermediate steps in $(8.2)$, but his assertions were all correct.