Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants? 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?
 A: 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.
A: As Frits Beukers write in http://www.staff.science.uu.nl/~beuke106/caen.pdf "Ironically all generalisations tried so far did not give any new interesting results. Only through a combination of miracles such generalisations seem to work, which in practice means that we fall back to $\zeta(2)$ or $\zeta(3)$ again".
Nevertheless Apéry-like numbers do appear in somewhat unexpected places:


*

*In the study of the  moments
$$W_n(s)=\int\limits_{[0,1]^n}\left |\sum_{k=1}^ne^{2\pi i x_k}\right|d\vec{x}$$
of  the  distance  traveled  by  a  walk  in  the  plane with unit steps in random directions. Namely for 3- and 4-step short random walks we have (see https://link.springer.com/article/10.1007/s11139-011-9325-y )
$$W_3(2k)=\sum_{j=0}^k\binom{k}{k}^2\binom{2j}{j},$$ and
$$W_4(2k)=\sum_{j=0}^k\binom{k}{k}^2\binom{2j}{j}\binom{2(k-j)}{k-j}.$$
A striking feature that connects the 3- and 4-step random walk densities to irrationality proofs is their modularity: https://arxiv.org/abs/1103.2995 (Densities of short uniform random walks, by J.M. Borwein, A> Straub, J. Wan and W. Zudilin). See Beuker's "Irrationality Proofs Using Modular Forms": http://carmasite.newcastle.edu.au/wadim/zw/Beukers-Asterisque1987.pdf

*Positivity of rational functions and their diagonals (an article by Armin Straub and Wadim Zudilin): https://arxiv.org/abs/1312.3732

*In some series for $1/\pi$:https://icerm.brown.edu/materials/Slides/tw-14-5/Apery_numbers_and_their_experimental_siblings_]_Armin_Straub,_University_of_Illinois_at_Urbana-Champaign.pdf

*Calabi-Yau differential equations: https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/generalizations-of-clausens-formula-and-algebraic-transformations-of-calabiyau-differential-equations/D5A27594AC2F57F9B933A7716506C027 (Generalizations of Clausen's Formula and algebraic transformations of Calabi–Yau differential equations, by
G. Almkvist, D. van Straten  and W. Zudilin).  
