On some supercongruence conjectures for truncated hypergeometric series. (English) Zbl 1482.11003
The authors determine many truncated hypergeometric series connected to the values of \(p\)-adic gamma function \(\Gamma_p (x)\), some of which confirming the Ramanujan-type supercongruences conjectured by L. van Hamme [Lect. Notes Pure Appl. Math. 192, 223–236 (1997; Zbl 0895.11051)] and investigated by H. Swisher [Res. Math. Sci. 2, Paper No. 18, 21 p. (2015; Zbl 1337.33005)].
Among the noteworthy results, the supercongruence hypothesized by J.-C. Liu [J. Math. Anal. Appl. 471, No. 1–2, 613–622 (2019; Zbl 1423.11015)] \[\sum_{k=0}^{\frac{p-1}{2}} (4k+1) (-1)^k \frac{\bigl (\frac{1}{2}\big )_{k}^{5}}{k!^5} \equiv -\frac{p^3}{16} \Gamma_p \Bigl (\frac{1}{4}\Big )^4 \pmod{p^5}\]is here established for any prime \(p \geq 5\) such that \(p \equiv 3 \pmod{4}\), whereas, on the basis of “numerical evidence”, \[\sum_{k=0}^{\frac{p^r-1}{2}} (4k+1) (-1)^k \frac{\bigl (\frac{1}{2}\big )_{k}^{5}}{k!^5} \equiv p^{2r} \pmod{p^{r+4}}\] is proposed as conjecture for positive even integers \(r\) and primes \(p>5\) with \(p \equiv 3 \pmod{4}\).
This paper also provides an alternative to the proof of V. J. W. Guo and S.-D. Wang [Proc. R. Soc. Edinb., Sect. A, Math. 150, No. 3, 1127–1138 (2020; Zbl 1468.11067)] for the van Hamme type supercongruence \[\sum_{k=0}^{\frac{p^r-1}{2}} (4k+1) \frac{\bigl (\frac{1}{2}\big )^4}{k!^4} \equiv p^{r} \pmod{p^{r+3}},\] valid for any prime \(p>3\) and any integer \(r>1\).
Beyond a proving method and a \(p\)-adic approximation to \(\Gamma_p\)-quotients both due to L. Long and R. Ramakrishna [Adv. Math. 290, 773–808 (2016; Zbl 1336.33018)], the authors employ a congruence about rising factorials they found in a previous work [Integral Transforms Spec. Funct. 30, No. 9, 683–692 (2019; Zbl 1439.11012)].
Among the noteworthy results, the supercongruence hypothesized by J.-C. Liu [J. Math. Anal. Appl. 471, No. 1–2, 613–622 (2019; Zbl 1423.11015)] \[\sum_{k=0}^{\frac{p-1}{2}} (4k+1) (-1)^k \frac{\bigl (\frac{1}{2}\big )_{k}^{5}}{k!^5} \equiv -\frac{p^3}{16} \Gamma_p \Bigl (\frac{1}{4}\Big )^4 \pmod{p^5}\]is here established for any prime \(p \geq 5\) such that \(p \equiv 3 \pmod{4}\), whereas, on the basis of “numerical evidence”, \[\sum_{k=0}^{\frac{p^r-1}{2}} (4k+1) (-1)^k \frac{\bigl (\frac{1}{2}\big )_{k}^{5}}{k!^5} \equiv p^{2r} \pmod{p^{r+4}}\] is proposed as conjecture for positive even integers \(r\) and primes \(p>5\) with \(p \equiv 3 \pmod{4}\).
This paper also provides an alternative to the proof of V. J. W. Guo and S.-D. Wang [Proc. R. Soc. Edinb., Sect. A, Math. 150, No. 3, 1127–1138 (2020; Zbl 1468.11067)] for the van Hamme type supercongruence \[\sum_{k=0}^{\frac{p^r-1}{2}} (4k+1) \frac{\bigl (\frac{1}{2}\big )^4}{k!^4} \equiv p^{r} \pmod{p^{r+3}},\] valid for any prime \(p>3\) and any integer \(r>1\).
Beyond a proving method and a \(p\)-adic approximation to \(\Gamma_p\)-quotients both due to L. Long and R. Ramakrishna [Adv. Math. 290, 773–808 (2016; Zbl 1336.33018)], the authors employ a congruence about rising factorials they found in a previous work [Integral Transforms Spec. Funct. 30, No. 9, 683–692 (2019; Zbl 1439.11012)].
Reviewer: Enzo Bonacci (Latina)
MSC:
| 11A07 | Congruences; primitive roots; residue systems |
| 11B65 | Binomial coefficients; factorials; \(q\)-identities |
| 33B15 | Gamma, beta and polygamma functions |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
Citations:
Zbl 0895.11051; Zbl 1337.33005; Zbl 1423.11015; Zbl 1468.11067; Zbl 1336.33018; Zbl 1439.11012Software:
SIGMAReferences:
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