A family of q -hypergeometric congruences modulo the fourth power of a cyclotomic polynomial
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A FAMILY OF q-HYPERGEOMETRIC CONGRUENCES MODULO THE FOURTH POWER OF A CYCLOTOMIC POLYNOMIAL
BY
Victor J. W. Guo∗ School of Mathematics and Statistics, Huaiyin Normal University Huai’an 223300, Jiangsu, People’s Republic of China e-mail: [email protected]
AND
Michael J. Schlosser∗∗ Fakult¨ at f¨ ur Mathematik, Universit¨ at Wien Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria e-mail: [email protected]
ABSTRACT
We prove a two-parameter family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial. Crucial ingredients in our proof are George Andrews’ multiseries extension of the Watson transformation, and a Karlsson–Minton-type summation for very-well-poised basic hypergeometric series due to George Gasper. The new family of q-congruences is then used to prove two conjectures posed earlier by the authors.
∗ The first author was partially supported by the National Natural Science Foun-
dation of China (grant 11771175).
∗∗ The second author was partially supported by FWF Austrian Science Fund grant
P 32305. Received September 23, 2019 and in revised form October 14, 2019
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V. J. W. GUO AND M. J. SCHLOSSER
Isr. J. Math.
1. Introduction In 1914, Ramanujan [26] presented a number of fast approximations of 1/π, including ∞
(1.1)
k=0
(6k + 1)
( 12 )3k 4 = , 3 k k! 4 π
where (a)n = a(a + 1) · · · (a + n − 1) denotes the rising factorial. In 1997, Van Hamme [31] proposed 13 interesting p-adic analogues of Ramanujan-type formulas, such as
(p−1)/2
(1.2)
k=0
(6k + 1)
( 12 )3k ≡ p(−1)(p−1)/2 k!3 4k
(mod p4 ),
where p > 3 is a prime. Van Hamme’s supercongruence (1.2) was first proved by Long [21]. It should be pointed out that all of the 13 supercongruences have been proved by different techniques (see [25,29]). For some background on Ramanujan-type supercongruences, the reader is referred to Zudilin’s paper [33]. In 2016, Long and Ramakrishna [22, Thm. 2] proved the following supercongruence: ⎧ p−1 ⎨−pΓp ( 1 )9 (mod p6 ), if p ≡ 1 (mod 6), ( 13 )6k 3 (1.3) (6k + 1) 6 ≡ ⎩− 10p4 Γ ( 1 )9 (mod p6 ), if p ≡ 5 (mod 6), k! k=0 p 3 27 where Γp (x) is the p-adic Gamma function. This result for p ≡ 1 (mod 6) confirms the (D.2) supercongruence of Van Hamme, which asserts a congruence modulo p4 . During the past few years, many congruences and supercongruences were generalized to the q-setting by various authors (see, for instance, [4–18, 23, 24, 28,30]). In particular, the authors [15, Thm. 2.3] proposed the following partial q-analogue of Long and Ramakrishna’s supercongruence (1.3): ⎧ n−1 ⎨0 (mod [n]), 3 6 if n ≡ 1 (mod 3), (q; q ) (1.4) [6k + 1] 3 3 k6 q 3k ≡ ⎩0 (mod [n]Φ (q)), if n ≡ 2 (mod 3). (q ; q )k k=0
n
Here and throughout the paper, we adopt the standard q-notation (cf. [3]): For an indeterminate q, let (a; q)n = (1 − a)(1 − aq) · · · (1 − aq n−1 )
q-HYPERGEOMETRIC CONGRUENCES
Vol. TBD, 2020
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be the q-shifted factorial. For convenience, we compactly write (a1 , a2 , . . . , am ; q)n = (a1 ; q)n (a2 ; q)n · · · (am ; q)n . Moreover, [n] = [n]q = 1
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