Publication
A Common q-Analogue of Two Supercongruences
Publisher:
Springer Science and Business Media LLC
Date:
09-03-2020
DOI:
10.1007/S00025-020-1168-7
Abstract: We give a q -congruence whose specializations $$q=-1$$ q = - 1 and $$q=1$$ 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): $$\\begin{aligned}& \\sum _{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\\equiv p(-1)^{(p-1)/2}\\ ({\\text {mod}}p^3) \\quad \\text {and}\\quad \\\\& \\sum _{k=0}^{(p-1)/2}A_k\\equiv a(p)\\ ({\\text {mod}}p^2), \\end{aligned}$$ ∑ k = 0 ( p - 1 ) / 2 ( - 1 ) k ( 4 k + 1 ) A k ≡ p ( - 1 ) ( p - 1 ) / 2 ( mod p 3 ) and ∑ k = 0 ( p - 1 ) / 2 A k ≡ a ( p ) ( mod p 2 ) , where $$p $$ p 2 is prime, $$\\begin{aligned} A_k=\\prod _{j=0}^{k-1}\\biggl (\\frac{1/2+j}{1+j}\\biggr )^3=\\frac{1}{2^{6k}}{\\left( {\\begin{array}{c}2k\\\\ k\\end{array}}\\right) }^3 \\quad \\text {for}\\ k=0,1,2,\\ldots , \\end{aligned}$$ A k = ∏ j = 0 k - 1 ( 1 / 2 + j 1 + j ) 3 = 1 2 6 k 2 k k 3 for k = 0 , 1 , 2 , … , and a ( p ) is the p th coefficient of the modular form $$q\\prod _{j=1}^\\infty (1-q^{4j})^6$$ q ∏ j = 1 ∞ ( 1 - q 4 j ) 6 (of weight 3). We complement our result with a general common q -congruence for related hypergeometric sums.