Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case that such new Theorems, whose substance contains completely novel approaches to an old problem, inspire solutions to other, similar problems by providing the tools necessary to tackle them.
However, in the case of Apéry's Theorem, it seems like the new techniques presented in the proof are not easily used within other contexts. This has led me to ask the question:
Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?
Since the proof contains numerous new ideas, I also put forward the following corollary question:
Which techniques employed in Apéry's proof of the irrationality of $\zeta(3)$ have been used within other proofs, whether in the field of irrationality/transcendence theory or other fields?
Best Answer
Regarding your second question, Apéry's amazing formula $$\zeta(3) = {5\over 2} \sum_{n\ge1} {(-1)^{n-1} \over n^3 {2n \choose n}}$$ has inspired the search for analogous formulas for other zeta function values. I think that the earliest such was conjectured by Borwein and Bailey and proved by Almkvist and Granville: $$\sum_{k\ge0} \zeta(4k+3) z^{4k} = {5\over 2} \sum_{n=1}^\infty {(-1)^{n-1}\over n^3 {2n\choose n}} {1 \over 1-z^4\!/n^4} \prod_{m=1}^{n-1} {1 + 4z^4\!/m^4 \over 1 - z^4\!/m^4}.$$ There are other results along these lines; you can search for "Apéry-like" to locate the relevant literature. Most of these formulas were found empirically. Unfortunately, I don't think that any of these formulas have led to actual irrationality proofs. Apéry's original series converges fast enough to enable an irrationality proof, but the more complicated formulas that were found later have behavior that is not so easy to analyze.