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.?


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