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Keyword: basic hypergeometric series ×Clear all

1 claim from 1 paper

1. Mathematics › Advanced Mathematical Identities

   A q-microscope for supercongruences

   Guo and Zudilin · Advances in Mathematics · 2019

   The paper proves polynomial congruences for truncated q-hypergeometric sums by studying their behaviour at roots of unity, yielding supercongruences for truncated ordinary hypergeometric sums.

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   1. Supported · 71%Two supercongruences for truncated sums of a Ramanujan series, modulo p cubed, are derived from a q-analogue and said to hold for all primes p above 3.“A typical example includes derivation, from a $q$-analogue of Ramanujan's formula $$ \sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1) =\frac{2\sqrt{3}}π, $$ of the two supercongruences $$ S(p-1)\equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3} \quad\text{and}\quad S\Bigl(\frac…”