New $q$-supercongruences arising from a summation of basic hypergeometric series

With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences. Especially, we give a $q$-analogue of a formula due to Liu [J. Math. Anal. Appl. 497 (2021), Art.~124915].

Noting N is a dense subset of Z p related to the p-adic norm | · | p , for each x ∈ Z p , the definition of p-adic Gamma function can be extended as Two properties of the p-adic Gamma function in common use can be stated as follows: The corresponding author * . Email addresses: weichuanan78@163.com (C. Wei), lichun@hainnu. edu.cn (C. Li).
where x p indicates the least nonnegative residue of x modulo p, i.e., x p ≡ x (mod p) and x p ∈ {0, 1, . . . , p − 1}. In 2016, Long and Ramakrishna [14,Proposition 25] showed that, for any prime p, For any complex numbers x and q, define the q-shifted factorial to be For simplicity, we also adopt the compact notation (x 1 , x 2 , . . . , x m ; q) n = (x 1 ; q) n (x 2 ; q) n · · · (x m ; q) n .
Let [n] = (1 − q n )/(1 − q) be the q-integer and Φ n (q) the n-th cyclotomic polynomial in q: where ζ is an n-th primitive root of unity. Motivated by the work just mentioned, we shall establish the following two theorems. Theorem 1.1. Let n be a positive integer with n ≡ 1 (mod 6). Then, modulo Φ n (q) 3 , Theorem 1.2. Let n be a positive integer with n ≡ 5 (mod 6). Then, modulo Φ n (q) 3 , It is not difficult to understand that Theorems 1.1 and 1.2 give a q-analogue of (1.2). Letting n = p be an prime and taking q → 1 in the above two theorems, we obtain the following conclusions. Corollary 1.3. Let p be an prime such that p ≡ 1 (mod 6). Then Corollary 1.4. Let p be an prime such that p ≡ 5 (mod 6). Then In order to explain the equivalence of (1.2) and Corollaries 1.3 and 1.4, we need to verify the following relations. Proposition 1.5. Let p be a prime such that p ≡ 1 (mod 6). Then Proposition 1.6. Let p be a prime such that p ≡ 5 (mod 6). Then The rest of the paper is arranged as follows. The proof of Theorems 1.1 and 1.2 will be given in Section 2. To this end, we first derive a q-supercongruence modulo (1−aq tn )(a−q tn )(b−q tn ), where t ∈ {1, 2}, by using a summation of basic hypergeometric series, the creative microscoping method, and the Chinese remainder theorem for coprime polynomials. Finally, the proof of Propositions 1.5 and 1.6 will be displayed in Section 3.
2 Proof of Theorems 1.1 and 1.2 In order to prove Theorems 1.1 and 1.2, we require the following lemma.
Proof. By comparing the k-th summands in the summations, it is easy to see that Evaluating the two series on the right-hand side by q-Saalschütz identity (cf. [1, Appendix (II.12)]): where Similarly, it is also routine to confirm the relation Calculating the two series on the right-hand side via (2.1), we arrive at Letting c → q −1 , x → ∞, y → ∞ in the last equation, we are led to Lemma 2.1.
Subsequently, we shall deduce the following united parametric extension of Theorems 1.1 and 1.2.
Proof. When a = q −tn or a = q tn , the left-hand side of (2.2) is equal to According to Lemma 2.1, the right-hand side of (2.3) can be written as Since (1 − aq tn ) and (a − q tn ) are relatively prime polynomials, we have the following result: modulo (1 − aq tn )(a − q tn ), When b = q tn , the left-hand side of (2.2) is equal to (2.5) By Lemma 2.1, the right-hand side of (2.5) can be expressed as Then we obtain the conclusion: modulo (b − q tn ), It is clear that the polynomials (1 − aq tn )(a − q tn ) and (b − q tn ) are relatively prime. Noting the q-congruences and employing the Chinese remainder theorem for coprime polynomials, we get Theorem 2.2 from (2.4) and (2.6).
Proof of Theorem 1.1. Letting b → 1, t = 2 in Theorem 2.2, we arrive at the formula: modulo Φ n (q)(1 − aq 2n )(a − q 2n ), By the L'Hôspital rule, we have Letting a → 1 in (2.7) and utilizing the above limit, we are led to the q-supercongruence: This completes the proof of Theorem 1.1.
Proof of Theorem 1.2. Letting b → 1, t = 1 in Theorem 2.2, we obtain the result: modulo By the L'Hôspital rule, we have Letting a → 1 in (2.8) and employing the upper limit, we get the q-supercongruence: modulo Φ n (q) 3 , Thus we finish the proof of Theorem 1.2. Moreover, it is not difficult to understand that Then we can proceed as follows: (1) 2 (p−2)/3