Abstract
We give a q-congruence whose specializations q=-1 and q=1 correspond to supercongruences (B.2) and (H.2) on Van Hamme’s list (in: p-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, vol 192. Dekker, New York, pp 223–236, 1997): ∑k=0(p-1)/2(-1)k(4k+1)Ak≡p(-1)(p-1)/2(modp3)and∑k=0(p-1)/2Ak≡a(p)(modp2),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}&\\sum _{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\\equiv p(-1)^{(p-1)/2}\\;({\\text {mod}}p^3) \\quad \\text {and}\\quad \\\\&\\sum _{k=0}^{(p-1)/2}A_k\\equiv a(p)\\;({\\text {mod}}p^2), \\end{aligned}$$\\end{document}where p>2 is prime, Ak=∏j=0k-1(1/2+j1+j)3=126k2kk3fork=0,1,2,…,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} A_k=\\prod _{j=0}^{k-1}\\biggl (\\frac{1/2+j}{1+j}\\biggr )^3=\\frac{1}{2^{6k}}{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) }^3 \\quad \\text {for}\\; k=0,1,2,\\ldots , \\end{aligned}$$\\end{document}and a(p) is the pth coefficient of the modular form qprod _{j=1}^infty (1-q^{4j})^6 (of weight 3). We complement our result with a general common q-congruence for related hypergeometric sums.
Highlights
The formula of Bauer [1] from 1859,∞ (−1)k(4k + 1)Ak = π2, k=01 2k 3 where Ak = 26k k for k = 0, 1, 2, . . . , (1.1)is one of traditional targets for different methods of proofs of hypergeometric identities
Its special status is probably linked to the fact that it belongs to a family of series for 1/π of Ramanujan type, after Ramanujan [21] brought to life in 1914 a long list of similar looking equalities for the constant but with a faster convergence
The method of creative microscoping used in our proofs indicates the origin of q-congruences from infinite q-hypergeometric identities; for example, the q-congruence (1.7) corresponds to the identity
Summary
Is one of traditional targets for different methods of proofs of hypergeometric identities. One notable—computer—proof of (1.1) was given in 1994 by Ekhad and Zeilberger [2] using the Wilf–Zeilberger (WZ) method of creative telescoping. It was observed in 1997 by Van Hamme [28] that many Ramanujan’s and Ramanujan-like evaluations have nice p-adic analogues; for example, the congruence (−1)k(4k + 1)Ak ≡ p(−1)(p−1)/2 (mod p3). The congruence (1.2) was first proved by Mortenson [19] using a 6F5 hypergeometric transformation; it later received another proof by one of these authors [29] via the WZ method [using the very same ‘WZ certificate’ as in [2] for (1.1)]. Hamme observed and proved (1.3) in [28], and it was later generalized by Sun [23,24, Theorem 2.5], Guo and Zeng [12, Corollary 1.2], Long and Ramakrishna [17], Liu [15,16, Theorem 1.5] in different ways.
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