Some q-supercongruences modulo the square and cube of a cyclotomic polynomial

Two q-supercongruences of truncated basic hypergeometric series containing two free parameters are established by employing specific identities for basic hypergeometric series. The results partly extend two q-supercongruences that were earlier conjectured by the same authors and involve q-supercongruences modulo the square and the cube of a cyclotomic polynomial. One of the newly proved q-supercongruences is even conjectured to hold modulo the fourth power of a cyclotomic polynomial.

We should point out that the q-congruence (1.3) does not hold for d = 3. The present authors [9] also established the following companion of (1.3): for any odd integer d ≥ 3 and integer n > 1, They also proposed the following conjectures [9, Conjectures 1 and 2], which are generalizations of (1.3) and (1.4).

Conjecture 1 Let d ≥ 5 be an odd integer. Then
Conjecture 2 Let d ≥ 5 be an odd integer and let n > 1. Then q-Supercongruences such as those above (modulo a third and even fourth power of a cyclotomic polynomial) are rather special. In fact, concrete results for truncated basic hypergeometric sums being congruent to 0 modulo a high power of a cyclotomic polynomial are very rare. See [8,[10][11][12]14,18] for recent papers featuring such results. The main goal of this paper is to add two complete two-parameter families of q-supercongruences to the list of such q-supercongruences (see Theorems 1 and 2).
We shall prove that the respective first cases of Conjectures 1 and 2 are true by establishing the following more general result.
Theorem 1 Let d and r be odd integers satisfying d ≥ 3, r ≤ d − 4 (in particular, r may be negative) and gcd(d, r ) = 1. Let n be an integer such that n ≥ d − r and n ≡ −r (mod d).
We shall also prove the following q-supercongruences.
The following generalization of the respective second cases of Conjectures 1 and 2 should be true.
We shall prove Theorems 1 and 2 in Sections 2 and 3, respectively, by making use of Andrews' multiseries extension (2.2) of the Watson transformation [1,Theorem 4], along with Gasper's very-well-poised Karlsson-Minton type summation [3,Eq. (5.13)]. It should be pointed out that Andrews' transformation plays an important part in combinatorics and number theory (see [7] and the introduction of [12] for more such examples).

Proof of Theorem 1
We need a simple q-congruence modulo n (q) 2 , which was already used in [10,12].

Lemma 1 Let α, r be integers and n a positive integer. Then
We will further utilize a powerful transformation formula due to Andrews [1,Theorem 4], which may be stated as follows: This transformation is a multiseries generalization of Watson's 8 φ 7 transformation formula (listed in [4, Appendix (III.18)]; cf. [4, Chapter 1] for the notation of a basic hypergeometric r φ s series we are using), (aq/be j , bq/e j ; q) n j (aq/e j , q/e j ; q) n j , (2.4) where n 1 , . . . , n m are non-negative integers, ν = n 1 + · · · + n m , and the convergence condition |q 1−ν /d| < 1 is required if the series does not terminate. We point out that an elliptic extension of the terminating d = q −ν case of (2.4) can be found in [26,Eq. (1.7)].
In particular, we note that for d = bq the right-hand side of (2.4) vanishes. Putting in addition b = q −N we get the following terminating summation formula: which is valid for N > ν = n 1 + · · · + n m . A suitable combination of (2.2) and (2.5) yields the following multi-series summation formula, derived in [12, Lemma 2] (whose proof we nevertheless give here, to make the paper self-contained): Lemma 2 Let m ≥ 2. Let q, a and e 1 , . . . , e m+1 be arbitrary parameters with e m+1 = e 1 , and let n 1 , . . . , n m and N be non-negative integers such that N > n 1 + · · · + n m . Then (2.6) Proof By specializing the parameters in the multi-sum transformation (2.2) by b i → aq n i +1 /e i , c i → e i+1 , for 1 ≤ i ≤ m (where e m+1 = e 1 ), and dividing both sides of the identity by the prefactor of the multi-sum, we obtain that the series on the right-hand side of (2.6) equals where 0 ≤ m ≤ n − 1 and dm ≡ −r (mod n).
We have collected enough ingredients which enables us to prove Theorem 1.
In what follows, we shall prove the modulus n (q) 3 case of (1.5).
Here, the q r , . . . , q r in the numerator means d − 1 instances of q r , and similarly, the q d , . . . , q d in the denominator means d − 1 instances of q d . By Andrews' transformation (2.2), we may rewrite the above expression as where m = (d + 1)/2. It is easy to see that the q-shifted factorial (q d+r ; q d ) (dn−n−r )/d contains the factor 1 − q (d−1)n which is a multiple of 1 − q n . Moreover, since none of (r − d)/2, (d + r )/2 and (d + r )/2 + dn − n − r − d are multiples of n, the q-shifted factorials (q (r −d)/2−(d−1)n ; q d ) (

dn−n−r )/d and (q (d+r )/2 ; q d ) (dn−n−r )/d
have the same number (0 or 1) of factors of the form 1 − q αn (α ∈ Z). Besides, the q-shifted factorial (q r −(d−1)n ; q d ) (dn−n−r )/d is relatively prime to n (q). Thus we conclude that the fraction before the multi-sum in (2.7) is congruent to 0 modulo n (q).
Note that the non-zero terms in the multi-summation in (2.7) are those indexed by  ( j 1 , . . . , j m−1 ) that satisfy the inequality j 1 + · · · + j m−1 ≤ (dn − n − r )/d because the factor (q r −(d−1)n ; q d ) j 1 +···+ j m−1 appears in the numerator. None of the factors appearing in the denominator of the multi-sum of (2.7) contain a factor of the form 1 − q αn (and are therefore relatively prime to n (q)), except for (q (3d+r )/2 ; q d ) j 1 +···+ j m−1 when the denominator of the above fraction contains a factor of the form 1 − q αn if and only if j 1 + · · · + j m−1 = (dn − d − n − r )/(2d) (in this case, the denominator contains the factor 1 − q (d−1)n/2 ). Writing n = ad − r (with a ≥ 1), we have j 1 + · · · + j m−1 = a(d − 1)/2 − (r + 1)/2. Noticing that m − 1 = (d − 1)/2 and r ≤ d − 4, there must exist an i such that j i ≥ a. Then (q d−r ; q d ) j i has the factor 1 − q d−r +d(a−1) = 1 − q n which is divisible by n (q). Hence the denominator of the reduced form of the multi-sum in (2.7) is relatively prime to n (q). It remains to show that the multi-sum in (2.7), without the previous fraction, is congruent to 0 modulo n (q) 2 .

Proof of Theorem 2
We first give a simple lemma on a property of certain arithmetic progressions.
Proof By the condition gcd(d, r ) = 1, we have gcd((d + r )/2, (d − r )/2) = 1. Suppose that for some integers a and b with a ≥ 0. Then thus implying that no number in the arithmetic progression (3.1) is a multiple of n.

Proof of Theorem 2
As before, the q-congruence (1.6) modulo [n] can be deduced from Lemma 3. It remains to prove the modulus n (q) 2 case of (1.6). For M = (dn − 2n − r )/d, the left-hand side of (1.6) can be written as the following multiple of a terminating d+5 φ d+4 series (this time we changed the position of q (d+r )/2 ): Here, the q r , . . . , q r in the numerator stands for d − 1 instances of q r , and similarly, the q d , . . . , q d in the denominator stands for d − 1 instances of q d . By Andrews' transformation (2.2), we may rewrite the above expression as where m = (d + 1)/2. It is easily seen that the q-shifted factorial (q d+r ; q d ) (dn−2n−r )/d has the factor 1−q (d−2)n which is a multiple of 1 − q n . Clearly, the q-shifted factorial (q −(d−2)n ; q d ) (dn−2n−r )/d has the factor 1 − q −(d−1)n (again being a multiple of 1 − q n ) since (dn − 2n − r )/d ≥ 1 holds according to the conditions d ≥ 3, r ≤ d − 4, and n ≥ (d − r )/2. This indicates that the q-factorial (q d+r , q −(d−2)n ; q d ) (dn−2n−r )/d in the numerator of the fraction before the multi-sum in (3.3) is divisible by n (q) 2 . Further, it is not difficult to see that the q-factorial (q d , q r −(d−2)n ; q d ) (dn−2n−r )/d in the denominator is relatively prime to n (q).
Like the proof of Theorem 1, the non-zero terms in the multi-sum in (3.3) are those indexed by ( j 1 , . . . , j m−1 ) satisfying the inequality j 1 + · · · + j m−1 ≤ (dn − 2n − r )/d because of the appearance of the factor (q r −(d−2)n ; q d ) j 1 +···+ j m−1 in the numerator. By Lemma 4, the q-shifted factorial (q (d+r )/2 , q d ) j 1 in the denominator does not contain a factor of the form 1 − q αn for j 1 ≤ (dn − 2n − r )/d (and are therefore relatively prime to n (q)). In addition, none of the other factors appearing in the denominator of the multisum of (3.3) contain a factor of the form 1 − q αn , except for (q d+r ; q d ) j 1 +···+ j m−1 when j 1 + · · · + j m−1 = (dn − 2n − r )/d (in this case the denominator contains the factor 1 − q (d−2)n ).
For M = n − 1, since (q r ; q d ) k /(q d ; q d ) k is congruent to 0 modulo n (q) for (dn − 2n − r )/d < k ≤ n − 1, we conclude that (1.6) is also true modulo n (q) 2 in this case.
Funding Open access funding provided by Austrian Science Fund (FWF).
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