Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?

Question

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?

Answer 1

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.

Answer 2

The proof of irrationality of $\displaystyle\sum_{n=0}^{+\infty}\frac1{F_n}$ (where $F_n$ is the $n$-th Fibonacci number) by RIchard André-Jeannin is an adaptation of the original Apery's proof of the irrationality of $\zeta(3)$.

Answer 3

As Frits Beukers writes 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:

1. In the study of the moments
   $$W_n(s)=\int\limits_{[0,1]^n}\left |\sum_{k=1}^ne^{2\pi i x_k}\right|^s d\vec{x}$$ 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 ) $$W_3(2k)=\sum_{j=0}^k\binom{k}{j}^2\binom{2j}{j},$$ and $$W_4(2k)=\sum_{j=0}^k\binom{k}{j}^2\binom{2j}{j}\binom{2(k-j)}{k-j}.$$ 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 Beukers's "Irrationality Proofs Using Modular Forms": http://carmasite.newcastle.edu.au/wadim/zw/Beukers-Asterisque1987.pdf

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

3. 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

4. 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).

Answer 4

I also want to mention the formula (origin unknown) $$\dfrac{56\zeta(3)}{3}=\sum_{n\ge1}\dfrac{64^n}{n^3D_nD_{n-1}}\;,$$ where $D_n=W_4(2n)$ is the $n$th Domb number, equivalent to the continued fraction $$\zeta(3)=12/7/(2-16/(36-1024/(160-11664/(434-\dots))))$$ (numerators $-16n^6$ denominators $(2n-1)(5n^2-5n+2)$), which converges essentially like $4^{-n}$, as well as the formula $$\dfrac{7\zeta(3)}{2}=\sum_{n\ge1}\dfrac{16^n}{n^3C_nC_{n-1}}\;,$$ where $C_n=\sum_{n/2\le k\le n}\binom{n}{k}^2\binom{2k}{n}^2$ (I do not know if these have a name), equivalent to the continued fraction $$\zeta(3)=8/7/(1-1/(21-64/(95-729/(259-4096/(549-\dots)))))$$ (numerators $-n^6$ denominators $(2n-1)(3n^2-3n+1)$), which converges essentially like $(1+\sqrt{2})^{-4n}$.