Hypergeometric evaluation identities and supercongruences

By Ling Long. 2009

176 citations recorded by the source corpus.

We apply some hypergeometric evaluation identities, including a strange valuation of Gosper, to prove several supercongruences related to special valuations of truncated hypergeometric series. In particular, we prove a conjecture of van Hamme.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/76b35f1c4a018f612e7f994da7d7e2ee29e1855e

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Citations in the corpus

Hypergeometric Functions over Finite Fields (2015, methodology): “For example, this technique is employed in work of the second, third, and fourth authors [62, 63, 85], among others.”
On the supercongruence conjectures of van Hamme (2015, methodology): “2) has been proved in three ways, by Mortenson [13] using a technical evaluation of a quotient of Gamma functions, by Zudilin [18] using the W-Z method, and by Long [9] using hypergeometric series identities and evaluations.”
Some congruences involving fourth powers of central q-binomial coefficients (2019, result): “As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]:”
A q-microscope for supercongruences (2018, methodology): “Development of methods [7, 20–22, 24, 25, 27, 31, 32, 34] for establishing Ramanujan-type supercongruences like (2) and (3), sometimes modulo a smaller power of p and normally on a case-by-case study, was mainly hypergeometric.”
Some supercongruences occurring in truncated hypergeometric series (2014, unknown): “Compared to [18] we provide a systematic way to carry out the p-adic analysis using the local analytic behavior of the p-adic Gamma function that handles the ordinary and supersingular cases equally well.”
Congruences concerning Legendre polynomials (2010, unknown): “2 p 4 (mod p2) for p ≡ 5,7 (mod 8) and p−1 X2 k=0 k 2k k 3 256k ≡ (−1)p−1 2 p 6 (mod p2) for p ≡ 2 (mod 3). Proof. From [M2] we know that p−1 X2 k=0 (4k +1) 2k k 3 (−64)k ≡ (−1)p−1 2 p (mod p3). From [L] we have p−1 X2 k=0 (6k +1) 2k k 3 256k ≡ (−1)p−1 2 p (mod p4). Thus applying Theorem 3.6 we dedu”
Super congruences and Euler numbers (2011, background): “≡p −1 p (mod p4) if p > 3, (p−X1)/2 k=0 (6k + 1) 2k k 3 (−512)k ≡p −2 p (mod p3), (pX−1)/2 k=0 (42k + 5) 2k k 3 4096k ≡5p −1 p (mod p4). The ﬁrst of these was recently shown by L. Long [Lo]; the second and the third remain open. Motivated by Theorem 1.3 and Lemma 4.1, we propose the followi”
Super congruences and Euler numbers (2010, background): “≡p −1 p (mod p4) if p > 3, (p−X1)/2 k=0 (6k + 1) 2k k 3 (−512)k ≡p −2 p (mod p3), (pX−1)/2 k=0 (42k + 5) 2k k 3 4096k ≡5p −1 p (mod p4). The ﬁrst of these was recently shown by L. Long [Lo]; the second and the third remain open. Motivated by Theorem 1.3 and Lemma 4.1, we propose the followi”
On the (K.2) supercongruence of Van Hamme (2015, methodology): “Long [6] utilized a combination of combinatorial identities, p-adic analysis and transformations together with “strange” evaluations of ordinary hypergeometric series due to Gosper, Gessel and Stanton to give yet another proof of (B.”
Supercongruences on Truncated Hypergeometric Series (2015, methodology): “3 by using the method of Long [[7], Section 3].”
On Van Hamme's (A.2) and (H.2) supercongruences (2018, methodology): “Now we solve this recurrence: In[3] := recSol = SolveRecurrence[rec, SUM[n]]”
A family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial (2019, unknown): “2) was first proved by Long [21].”
A refinement of a congruence result by van Hamme and Mortenson (2010, background): “k k 3 (−64)k ≡ p −1 p +p3E p−3 (mod p 4). (1.1) Remark1.1. The only previously proved congruence mod p4 of the same kind is the following one conjectured by van Hamme [vH] and conﬁrmed by L. Long [Lo]: (p−X1)/2 k=0 (6k +1) 2k k 3 256k ≡ p −1 p (mod p4) for any prime p > 3. A REFINEMENT OF A C”