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Supercongruences for truncated Appell series. (English) Zbl 1459.11010

After defining the truncated Appell series \(F_1\) and \(F_2\) and recalling the \(p\)-adic Gamma function \(\Gamma_p(x)\), this paper establishes that, for any prime \(p \geq 5 \), \[ F_1 \left[ \tfrac12; \tfrac12, \tfrac12; 1; 1, 1 \right] _ \frac{ p-1}{2} \equiv p \pmod {p^2},\] \[ F_2 \left[ \tfrac12; \tfrac12, \tfrac12; 1, 1; 1, 1 \right] _ \frac{ p-1}{2} \equiv \bigg\{ \begin {array}{rl} -\Gamma_p (\frac{1}{4})^4 \pmod {p^2} & \text{if } p \equiv 1 \pmod {4},\\ 0 \pmod {p^2} & \text{if } p \equiv 3 \pmod {4}.\\ \end {array}\]
Beyond basic tools like Wolstenholme’s theorem, Fermat’s Little Theorem, and the Chu-Vandermonde identity, the proof employs some congruences, involving harmonic numbers, given by E. Lehmer [Ann. Math. (2) 39, 350–360 (1938; Zbl 0019.00505)] and by R. Tauraso [Int. J. Number Theory 14, 1093–1109 (2018; Zbl 1421.11008)] and it also implements the \(mZ\) algorithm supplied by M. Apagodu and D. Zeilberger [Adv. Appl. Math. 37, No. 2, 139–152 (2006; Zbl 1108.05010)].
The author remarks the connection between his result on \(F_2\) and a supercongruence, conjectured by L. van Hamme [Lect. Notes Pure Appl. Math. 192, 223–236 (1997; Zbl 0895.11051)], about the truncated hypergeometric series \({_{3}F_2}\).

MSC:

11A07 Congruences; primitive roots; residue systems
33C65 Appell, Horn and Lauricella functions
33C20 Generalized hypergeometric series, \({}_pF_q\)
33B15 Gamma, beta and polygamma functions
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)
05A19 Combinatorial identities, bijective combinatorics

Software:

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References:

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