Creative microscoping

Speaker: 

Wadim Zudilin

Affiliation: 

Radboud University Nijmegen

Date: 

Wed, 21/08/2019 - 2:00pm

Venue: 

RC-4082, The Red Centre, UNSW

Abstract: 

Ramanujan's formulas for $1/\pi$ and their generalizations remain an amazing topic, with many mathematical challenges.
Recently it was observed that the formulas possess spectacular `supercongruence' counterparts.
For example, truncating the sum in Ramanujan's formula
$$
\sum_{k=0}^\infty\frac{\binom{4k}{2k}{\binom{2k}{k}}^2}{2^{8k}3^{2k}}\,(8k+1)=\frac{2\sqrt{3}}{\pi}
$$
to the first $p$ terms correspond to the congruence
$$
\sum_{k=0}^{p-1}\frac{\binom{4k}{2k}{\binom{2k}{k}}^2}{2^{8k}3^{2k}}\,(8k+1)\equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3}
$$
valid for any prime $p>3$.
Some supercongruences were shown earlier through a tricky use of classical hypergeometric identities or the Wilf-Zeilberger method of creative telescoping.
The particular example displayed above (and many other entries) were resistant to such techniques.
In joint work with Victor Guo we develop a new method of `creative microscoping' that simultaneously proves both the underlying Ramanujan's formula and its finite supercongruence counterparts.
The main ingredient is an asymptotic analysis of $q$-analogues of Ramanujan's formulas at roots of unity.

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