A New Family of q -Supercongruences Modulo the Fourth Power of a Cyclotomic Polynomial
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Results in Mathematics
A New Family of q-Supercongruences Modulo the Fourth Power of a Cyclotomic Polynomial Victor J. W. Guo and Michael J. Schlosser Abstract. We establish a new family of q-supercongruences modulo the fourth power of a cyclotomic polynomial, and give several related results. Our main ingredients are q-microscoping and the Chinese remainder theorem for polynomials. Mathematics Subject Classification. Primary 33D15, Secondary 11A07, 11B65. Keywords. Basic hypergeometric series, supercongruences, q-congruences, cyclotomic polynomial, q-microscoping, Chinese remainder theorem for polynomials.
1. Introduction More than one hundred years ago, Ramanujan mysteriously recorded a list of rapidly convergent series of 1/π (see [1, p. 352]), including √ ∞ 4n2n2 2 3 2n n , (1.1) (8n + 1) = 28n 32n π n=0 which he later published in [19, Equation (40)]. In 1997, Van Hamme [23] observed that 13 Ramanujan’s and Ramanujan-type formulas possess interesting p-adic analogues, such as Victor J. W. Guo was partially supported by the National Natural Science Foundation of China (Grant 11771175). Michael J. Schlosser was partially supported by FWF Austrian Science Fund Grant P32305. 0123456789().: V,-vol
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V. J. W. Guo and M. J. Schlosser
(p−1)/2
(4k + 1)
k=0
( 12 )4k ≡p k!4
Results Math
(mod p3 ),
(1.2)
where p > 3 is a prime and (a)n = a(a + 1) · · · (a + n − 1) is the Pochhammer symbol. Van Hamme [23, (C.2)] himself proved (1.2) and two of the other supercongruences of his list. The supercongruence (1.2) was later proved to be true modulo p4 by Long [17]. For more Ramanujan-type supercongruences, we refer the reader to Zudilin’s paper [26]. During the past few years, q-analogues of congruences and supercongruences have been investigated by many authors (see [3–16,18,21,22,24,25,27]). For instance, using a method similar to that used in [26], the first author and Wang [12, Theorem 1.2] gave a q-analogue of (1.2): for odd n,
(n−1)/2
[4k + 1]
k=0
(q; q 2 )4k (q 2 ; q 2 )4k
≡ q (1−n)/2 [n] +
(n2 − 1)(1 − q)2 (1−n)/2 3 q [n] 24
(mod [n]Φn (q)3 ).
(1.3)
Moreover, the first author and Zudilin [13] devised a method of ‘creative microscoping’ to prove that, for any positive integer n with gcd(n, 6) = 1,
(n−1)/2
(q; q 2 )2k (q; q 2 )2k 2k2 q (q 2 ; q 2 )2k (q 6 ; q 6 )2k k=0 −3 −(n−1)/2 ≡q [n] (mod [n]Φn (q)2 ), n [8k + 1]
(1.4)
where ( −3 · ) is the Jacobi symbol, see [13, Theorem 1.1, Equation (6)]. Here it is appropriate to recall the standard q-hypergeometric notation: (a; q)n = (1 − a)(1 − aq) · · · (1 − aq n−1 ) is the q-shifted factorial, with the condensed notation (a1 , . . . , am ; q)n = (a1 ; q)n · · · (am ; q)n for products of q-shifted factorials; [n] = [n]q = (1−q n )/(1−q) is the q-integer; and Φn (q) stands for the nth cyclotomic polynomial in q: (q − ζnk ), Φn (q) = 1kn gcd(k,n)=1
where ζn denotes an nth primitive root of unity. Clearly, the q-supercongruence (1.4) is a q-analogue of the following result (see [20, Conjecture 5.6]):
4k2k2
(p−1)
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