# Alternative approaches to Zudilin's proof of Apéry's theorem

The article

Wadim Zudilin, Apéry's theorem. Thirty years after, Int. J. Math. Comput. Sci. 4 (2009), no. 1 pp 9–19. (arXiv:math/0202159, with the title An elementary proof of Apéry's theorem)

introduces the functions $$R_n(t)=\displaystyle {\Big{(}} \frac{(t-1)..(t-n)}{t(t+1)..(t+n)} {\Big{)}}^2$$, where $$n \in \mathbb{N}^*$$ and $$t \in \mathbb R \setminus \mathbb Z_-$$.

Zudilin proves that $$A_n=B_n$$ for all $$n \in \mathbb{N}^*$$, where $$A_n=\sum_{t=1}^{+ \infty} R'_n(t)$$ and $$B_n=\sum_{t=1}^{+ \infty} \frac{(n!)^2 (2t+n)(t-1) \dots (t-n)(t+n+1) \dots (t+2n)}{(t(t+1) \dots (t+n))^4} \ .$$

To prove this he shows that $$A_n$$ and $$B_n$$ satisfy both a same explicit recurrence relation of order $$2$$ with same initial condition. Did anyone have an idea to prove that $$A_n=B_n$$ using any other method, for instance, integral representation of $$A_n$$, hypergeometric sum etc.?