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Hypergeometric evaluation identities and supercongruences. (English) Zbl 1215.33002

The author establishes the truth of the following (super)congruences for primes \(p>3\):
\[ \begin{aligned} \sum_{k=0}^{(p^r-1)/2}\frac{4k+1}{2^{8k}}{\binom{2k}k}^4 &\equiv p^r \pmod{p^{r+3}}, \\ \sum_{k=0}^{(p-1)/2}\frac{4k+1}{2^{12k}}{\binom{2k}k}^6 &\equiv p\cdot a_p \pmod{p^4}, \\ \sum_{k=0}^{(p-1)/2}\frac{6k+1}{2^{8k}}{\binom{2k}k}^3 &\equiv(-1)^{(p-1)/2}p \pmod{p^4}, \\ \sum_{k=0}^{(p-1)/2}\frac{6k+1}{(-1)^k2^{9k}}{\binom{2k}k}^3 &\equiv(-1)^{(p^2-1)/8+(p-1)/2}p \pmod{p^2}, \end{aligned} \] where \(r\) is an arbitrary positive integer and \(a_p\) denotes the \(p\)-th coefficient of the weight 4 cusp form \(q\prod_{n=1}^\infty(1-q^{2n})^4(1-q^{4n})^4\).
The proofs make use of ‘strange’ evaluations of hypergeometric series [I. Gessel and D. Stanton, SIAM J. Math. Anal. 13, 295–308 (1982; Zbl 0486.33003)].

MSC:

33C20 Generalized hypergeometric series, \({}_pF_q\)
11F33 Congruences for modular and \(p\)-adic modular forms
33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)

Citations:

Zbl 0486.33003
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