Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised 12ϕ11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{12}\phi _{11}$$\end{document} series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new 12ϕ11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_{12}\phi _{11}$$\end{document} transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.


Introduction
Ramanujan, in his second letter to Hardy on February 27, 1913, mentioned the following identity where (x) is the Gamma function and where (a) k = a(a + 1) · · · (a + k − 1) is the Pochhammer symbol. A p-adic analogue of (1.1) was conjectured by Van Hamme [55, Eq. (A. 2)] as follows: Here and throughout the paper, p always denotes an odd prime and p (x) is the p-adic Gamma function. The congruence (1.2) was later proved by McCarthy and Osburn [43] through a combination of ordinary and Gaussian hypergeometric series. Recently, the congruence (1.2) for p ≡ 3 (mod 4) and p > 3 was further generalized by Liu [37] to the modulus p 4 case. It is well known that some truncated hypergeometric series are closely related to Calabi-Yau threefolds over finite fields and are further relevant to the coefficients of modular forms. For example, using the fact that the Calabi-Yau threefold in question is modular, which was proved by Ahlgren and Ono [3], Kilbourn [34] succeeded in proving Van Hamme's (M.2) supercongruence: where a p is the pth coefficient of a weight 4 modular form (1 − q 2n ) 4 (1 − q 4n ) 4 , q = e 2πiz .
Applying Whipple's 7 F 6 transformation formula, Long [40] proved that The main aim of this paper is to give q-analogues of some known supercongruences, including a partial q-analogue of Long's supercongruence (1.4) (partial in the sense that the modulo p 4 condition is replaced by the weaker condition modulo p 3 ). We provide such a result in Theorem 2.1 in the form of two transformations of truncated basic hypergeometric series. In addition, several other q-supercongruences are given. These results are proved by special instances of transformation formulas for basic hypergeometric series. (See Theorem A.1 in the Appendix for a new basic hypergeometric transformation formula which we make use of.) Throughout we assume q to be fixed with 0 < |q| < 1. We refer to q as the "base". For a, k ∈ C, the q-shifted factorial is defined by (1 − aq j ). (1.5) For brevity, we frequently use the shorthand notation (a 1 , . . . , a m ; q) k = (a 1 ; q) k . . . (a m ; q) k , k ∈ C ∪ ∞.
Moreover, the q-binomial coefficients x k are defined by It is easy to see that (1.6) Following Gasper and Rahman [13], basic hypergeometric r φ s series with r upper parameters a 1 , . . . , a r , s lower parameters b 1 , . . . , b s , base q, and argument z are defined by where q = 0 when r > s + 1. Such a series terminates if one of the upper parameters, say, a r , is of the form q −n , where n is a nonnegative integer. If the series does not terminate, then it converges for |z| < 1.
In many of our proofs we will make use of Watson's 8 φ 7 transformation formula [13, Appendix (III. 17 which is valid whenever the 8 φ 7 series converges and the 4 φ 3 series terminates. In particular, we will also make use of the limiting case f = q −n → ∞, which we state for convenience: Other transformations we make use of are a quadratic transformation formula of Rahman, stated in (6.3), a cubic transformation formula of Gasper and Rahman, stated in (7.1), a quartic transformation formula by Gasper and Rahman, stated in (8.1), a double series transformation by Ismail, Rahman and Suslov, stated in (11.1), and a new transformation formula for a nonterminating 12 φ 11 series into two multiples of nonterminating 4 φ 3 series, given as Theorem A.1 in the Appendix. We also make use of the q-Dixon summation, stated in (10.1). For further material on basic hypergeometric series and more generally, on special functions, we refer to the text books by Gasper and Rahman [13], and by Andrews, Askey and Roy [2], respectively. In particular, in our computations we implicitly make heavy use of elementary manipulations of q-shifted factorials (see [13,Appendix I]).
Recall that the q-integer is defined as [n] = [n] q = 1 + q + · · · + q n−1 . Moreover, the nth cyclotomic polynomial n (q) is given by where ζ is an nth primitive root of unity. It is clear that n (q) is a polynomial in q with integer coefficients. Further, d|n, d>1 in particular, p (q) = [p] for any prime p.
We say that two rational functions A(q) and B(q) in q are congruent modulo a polynomial P(q), denoted by A(q) ≡ B(q) (mod P(q)), if the numerator of the reduced form of A(q) − B(q) is divisible by P(q) in the polynomial ring Z[q].

The Main Results
The following is our q-analogue of (1.4), where the modulo p 4 condition is replaced by the weaker condition modulo p 3 .

Theorem 2.1 Let n be a positive odd integer. Then
and Noticing that the terms corresponding to k in the upper half range (n−1)/2 < k ≤ n−1 are congruent to 0 modulo n (q) 3 (2. 2) The first author and Zeng [27, Cor. 1.2] gave a q-analogue of (2.2) as follows: We do not know any q-analogue of (1.2). However, we are able to provide a q-analogue of a very closely related congruence. In particular, since p ( 1 4 ) 4 p ( 3 4 ) 4 = 1, from (1.2) and (2.2) we deduce that which was already noticed by Mortenson [44]. We are able to give the following complete q-analogue of (2.3).

Theorem 2.2 Let n be a positive odd integer. Then
This result is stronger than Van Hamme's (D.2) supercongruence conjecture which asserts a congruence modulo p 4 for p ≡ 1 (mod 6). Long and Ramakrishna also pointed out that (2.5) does not hold modulo p 7 in general. We propose the following partial q-analogue of Long and Ramakrishna's supercongruence (2.5).

Theorem 2.3 Let n be a positive integer coprime with 3. Then
(2.6) We also partially confirm the a = 1 case of the second congruence in [28,Conj. 5.2].

Theorem 2.4 Let d and n be positive integers with d > 2 and n ≡ −1 (mod d). Then
The proofs of Theorems 2.3 and 2.4 are deferred to Sect. 4. In Sect. 3, we shall prove Theorems 2.1 and 2.2 using the creative microscoping method developed by the first author and Zudilin [28]. Roughly speaking, to prove a q-supercongruence modulo n (q) 3 , we prove its generalization with an extra parameter a so that the corresponding congruence holds modulo n (q)(1 − aq n )(a − q n ). Since the polynomials n (q), 1 − aq n , and a − q n are pairwise relatively prime, this generalized q-congruence can be established modulo these three polynomials individually. Finally, by taking the limit a → 1, we obtain the original q-supercongruence of interest. We learned that this creative microscoping method has already caught the interests of Guillera [14] and Straub [50].
Further, we introduce a new idea for proving some congruences modulo n (q). In many instances in this paper, the congruences (n−1)/2 k=0 a n,k ≡ 0 (mod n (q)) are proved by simply showing a k + a (n−1)/2−k ≡ 0 (mod n (q)) (instead of, say, evaluating certain infinite series at roots of unity which was illustrated in [28]).
The proofs of Theorems 2.3 and 2.4 in Sect. 4 again are done by showing a more general identity but otherwise are accomplished in a slightly different way. All the proofs of Theorems 2.1-2.4 in Sects. 3 and 4, and of the further results from Sect. 5, are based on Watson's 8 φ 7 transformation formula. We also confirm a three-parametric q-congruence conjecture in Sect. 6 based on a quadratic transformation formula of Rahman. Further, in Sect. 7 we deduce some q-congruences from a cubic transformation formula of Gasper and Rahman. Similarly, in Sect. 8 we deduce some q-congruences from a quartic transformation formula of Gasper and Rahman. The q-supercongruences in Sect. 9 are proved similarly but are derived using a new 12 φ 11 transformation formula. Since the latter formula is of independent interest, its derivation is given in the Appendix. It is also shown there how a special case of the 12 φ 11 transformation formula can be utilized to obtain a generalization of Rogers' linearization formula for the continuous q-ultraspherical polynomials. In Sect. 10 some q-supercongruences are deduced from the q-Dixon summation. In Sect. 11 we deduce q-super congruences-most of them only conjectural-from a double series transformation of Ismail, Rahman and Suslov. Finally, in Sect. 12, some concluding remarks are given and some related conjectures for further study are proposed. For example, we conjecture that the congruence (2.6) still holds modulo [n] n (q) 3 for n ≡ 2 (mod 3).

Proofs of Theorems 2.1 and 2.2
We first give the following lemma. Lemma 3.1 Let n be a positive odd integer. Then, for 0 ≤ k ≤ (n − 1)/2, we have Proof Since q n ≡ 1 (mod n (q)), we have (3.1) Further, modulo n (q), we have which in combination with (3.1) establishes the assertion.
We now use the above lemma to prove the following result which was originally conjectured by the first author and Zudilin [28,Conj. 5.6].
Noticing that q n ≡ 1 (mod n (q)) and, for odd n, n (q 2 ) = n (q) n (−q), we get for any positive integer n with n ≡ 3 (mod 4) and 0 ≤ k ≤ (n − 1)/2. This completes the proof of the theorem.
Similarly, we can prove that the third q-congruence in [28, Conj. 5.2] is true modulo n (q) and is therefore further true modulo [n] (again as in the proof of Theorem 2.1). We shall establish the following two-parameter generalization of Theorem 2.1.

Theorem 3.3 Let n be a positive odd integer. Then, modulo n
Proof For a = q −n or a = q n , the left-hand side of (3.2) is equal to By Watson's 8 φ 7 transformation formula (1.7), we can rewrite the right-hand side of (3.3) as It is easy to see that the fraction before the sum on the right-hand side of (3.4) is equal to [n]q (1−n)/2 . This proves that the congruence (3.2) holds modulo 1 − aq n or a − q n .
Moreover, by Lemma 3.1, it is easy to see that, modulo n (q), the kth and ((n − 1)/2 − k)th terms on the left-hand side of (3.2) cancel each other, i.e., [2n − 4k − 1](aq, q/a, bq; q 2 ) (n−1)/2−k (q; q 2 ) 3 When the left-hand side of (3.2) has an odd number of factors, the central term will remain. This happens when n = 4l + 1 for some positive integer l, and in this case the central term has index k = l and one directly sees that [4k + 1] = [n] is a factor of the summand. In total, this proves that the left-hand side of (3.2) is congruent to 0 modulo n (q), and therefore the congruence (3.2) also holds modulo n (q). Since n (q), 1 − aq n and a − q n are pairwise relatively prime polynomials, the proof of (3.2) is complete.

Proof of Theorem 2.1
The limits of the denominators on both sides of (3.2) as a → 1 are relatively prime to n (q), since 0 ≤ k ≤ (n − 1)/2. On the other hand, the limit of (1 − aq n )(a − q n ) as a → 1 has the factor n (q) 2 . Thus, the limiting case a, b → 1 of (3.2) gives the following congruence which also implies that since (q; q 2 ) 6 k /(q 2 ; q 2 ) 6 k ≡ 0 (mod n (q) 3 ) for k in the range (n − 1)/2 < k ≤ n − 1. It remains to show that the above two congruences are still true modulo [n], or equivalently,

7a)
and n−1 For n > 1, let ζ = 1 be an nth root of unity, not necessarily primitive. That is, ζ is a primitive root of unity of odd degree d | n. Let c q (k) denote the kth term on the left-hand side of the congruences in (3.7), i.e., The congruences (3.5) and (3.6) with n = d imply that Observe that We have which means that the sums n−1 k=0 c q (k) and (n−1)/2 k=0 c q (k) are both divisible by the cyclotomic polynomial d (q). Since this is true for any divisor d > 1 of n, we conclude that they are divisible by d|n, d>1 thus establishing (3.7).

Proof of Theorem 2.2
Similarly as in the proof of Theorem 2.1, letting a → 1 and b → ∞ in (3.2), we obtain which also implies that Along the same lines as in the proof of Theorem 2.1, we can show that Combining the above congruences, we are led to (2.4a) and (2.4b).

Proofs of Theorems 2.3 and 2.4
We shall prove the following common generalization of Theorems 2.3 and 2.4.

Theorem 4.1 Let n and d be positive integers with d
Proof Let α and j be integers. Since It follows that Similarly, we have Since gcd(n, d) = 1, we know that there exists a positive integer α < d such that αn ≡ 1 (mod d). Then by [13, Appendix (III.18)] (i.e., (1.7) with f = q 1−αn ), modulo n (q) 2 , the left-hand side of (4.1) is congruent to It is clear that (q d+1 ; q d ) (αn−1)/d in the numerator has the factor 1 − q αn and is therefore divisible by n (q), while the denominator is coprime with n (q). This proves (4.1) for the second case. Furthermore, if n ≡ −1 (mod d), then, modulo n (q) 2 , the left-hand side of (4.1) is congruent to It is easy to see that this time the numerator has the factor and is therefore divisible by n (q) 2 , and again the denominator is coprime with n (q). This proves (4.1) for the first case.
Letting a = 1 and d = 3 in (4.1), we get Similarly as in the proof of Theorem 2.1, we can prove that This completes the proof of (2.6). Likewise, taking a → 0 in (4.1), we obtain This completes the proof of (2.7). It appears that the following generalization with one more parameter b is still true.

More q-Congruences from Watson's Transformation
Throughout this section, m always stands for n − 1 or (n + 1)/2. Note that the special case of [28,Thm. 4.9] with r = −1, d = 2 and a = 1 gives In this section, we shall give some similar congruences.
Proof We first establish the following result: For a = q −n or a = q n , the left-hand side of (5.2) is equal to By the limiting case of Watson's transformation formula (1.8), we can rewrite the right-hand side of (5.3) as It is easy to see that This proves that the congruence (5.2) holds modulo On the other hand, by Lemma 3.1, for 0 ≤ k ≤ (n + 1)/2, we have It follows that the kth and ((n + 1)/2 − k)th terms on the left-hand side of (5.2) cancel each other modulo n (q). When the respective sum has an odd number of factors, the central term will remain. This happens when n = 4l −1 for some positive integer l, and in this case the central term has index k = l and one directly sees that is a factor of the summand. In total, this proves that the congruence (5.2) also holds modulo n (q). Since the polynomials n (q), 1 − aq n and a − q n are coprime with one another, the proof of (5.2) is complete.
Letting a → 1 in (5.2), one sees that the congruence (5.1) holds modulo n (q) 3 by noticing that (q −1 ; Along the same lines of the proof of Theorem 2.1, we can prove that i.e., the congruence (5.1) holds modulo [n]. Since lcm( n (q) 3 , [n]) = [n] n (q) 2 , the proof of the theorem is complete.

Theorem 5.3 Let n > 1 be an odd integer. Then
We first establish the following congruence: Letting q → q 2 and c → 0 followed by Letting q → q −1 in the above equality, we see that the summation on the righthand side of (5.7) is equal to 0. This proves that the congruence (5.6) holds modulo On the other hand, similarly as before, by (5.4) one sees that the sum of the kth and ((n + 1)/2 − k)th terms on the left-hand side of (5.6) are congruent to 0 modulo n (q) (and also, when the respective sum has an odd number of factors, i.e., when n = 4l − 1 for some positive integer l, then the remaining central term has index k = l and one directly sees that [4k − 1] = [n] is a factor of the summand). This thus proves that the congruence (5.6) is also true modulo n (q). This completes the proof of (5.6).
Let c q (k) denote the kth term on the left-hand side of (5.6). In the same vein as in the proof of Theorem 2.1, we can further prove that (5.10) The parts of the denominators in (5.10) which contain the parameter a are the factors of (aq 2 , q 2 /a; q 2 ) (n+1)/2 or (aq 2 , q 2 /a; q 2 ) n−1 . Their limits as a → 1 are relatively prime to n (q). On the other hand, the limit of (1 − aq n )(a − q n ) as a → 1 has the factor n (q) 2 . Therefore, the limiting case a → 1 of the congruence (5.10) reduces to (5.5) modulo n (q) 3 . But the congruences (5.9) are still true when a = 1 which implies that the congruence (5.5) holds modulo [n]. This completes the proof of the theorem.
It appears that the congruence conditions stated in Theorem 5.3 and its extension in (5.6) can be strengthened:

Theorem 5.5 Let n > 3 be an odd integer. Then
The proof is similar to that of Theorem 5.3. We first establish As we have already mentioned in the proof of Theorem 5.3, the summation on the right-hand side of (5.13) is equal to 0 by the q → q −1 case of (5.8). Thus, we have proved that the congruence (5.12) holds modulo (1 − aq n )(a − q n ).
On the other hand, similarly as before, by (5.4) one sees that the sum of the kth and ((n − 1)/2 − k)th terms on the left-hand side of (5.12) are congruent to 0 modulo n (q) for 0 ≤ k ≤ (n − 1)/2. Moreover, the summand for k = (n + 1)/2 on the right-hand side of (5.13) is clearly congruent to 0 modulo n (q) because of the factor (q; q 2 ) (n+1)/2 in the numerator. This proves that the congruence (5.12) is also true modulo n (q). The proof of (5.12) is completed.
For n > 3, we have (n + 3)/2 < n and so the denominator of the left-hand side of (5.12) is relatively prime to n (q) when taking the limit as a → 1. Therefore, the congruence (5.11) holds modulo n (q) 3 for n > 3 by taking a → 1 in (5.12). On the other hand, it is also easy to see that the congruence (5.11) holds modulo 3 (q) for n = 3. Let c q (k) denote the kth term on the left-hand side of (5.12). Similarly to the proof of Theorem 2.1, we can further prove that This proves (5.11).
We conjecture that the following generalization of (5.12) and Theorem 5.5 is still true.

Conjecture 5.6 Let n > 3 be an odd integer. Then
in particular, Analogously, letting q → q 2 and c → 0 followed by a = q, b = q −1 , d = q −1−n , and e = q −1+n in (1.8), we can prove the following result: We label the limiting case a → 1 as the following theorem.
It seems that the following generalization of (5.14) and (5.15) still holds.

Conjecture 5.8 Let n > 3 be a positive odd integer. Then
in particular, We also have the following similar result.

Theorem 5.9
Let n > 1 be a positive odd integer. Then Proof It is easy to see by induction on N that Putting N = (n − 1)/2 in the above identity and using (1.6), we get Let n = p and q = 1 in (5.17). Using Fermat's little theorem, we immediately obtain the following conclusion. Corollary 5. 10 We have We end this section with the following conjecture, which is similar to Conjecture 4.2. As in the proof of Theorem 4.1, we can confirm it for b = 1.

Conjecture 5.11
Let n and d be positive integers with d ≥ 3 and gcd(n, d) = 1. Then
The congruence (6.1) modulo (1 − aq n )(a − q n ) has already been proved by the first author and Zudilin [28,Thm. 4.7]. Moreover, the congruence (6.1) with c = 1 was established in [28,Thm. 4.8]. Therefore, it remains to prove that (6.1) holds modulo (aq, q/a, q; q 2 ) k (q/b, q/c, bc; q) k (aq, q/a, q; q) k (bq 2 , cq 2 , q 3 /bc; q 2 ) k q k ≡ 0 (mod [n]). (6.2) Proof We need to use a quadratic transformation formula of Rahman [47] (see also [13,Eq. (3.8.13)]): provided d or aq/d is not of the form q −2n , n a non-negative integer. It is clear that (6.2) is true for n = 1. We now suppose that n > 1. Let a = q 1−n in (6.3) and then we further set d = aq and replace b and c with q/b and q/c, respectively. Then the left-hand side of (6.3) terminates at k = (n − 1)/2, and the right-side of (6.3) vanishes because the numerator contains the factor (q 3−n ; q 2 ) ∞ . Namely, we have Since q n ≡ 1 (mod n (q)), we immediately get Finally, the proof of (6.2) is completely analogous to that of Theorem 2.1 (more precisely, to the proofs of (3.7a) and (3.7b)).

Conjecture 6.2 We have
Unfortunately, we were not able to find any q-analogue of (6.9), even for the simple case modulo p. Moreover, letting a → 1, b → −1, and c → 0 in (6.1), we get while letting a → 1 and b, c → 0 in (6.1), we arrive at It is worth mentioning that both (6.10) and (6.11) are q-analogues of the following supercongruence due to Guillera and Zudilin [15]: The congruence (6.11) with M = n − 1 was first established by the first author [21] using the q-WZ method. The congruence (6.10) is new.
This completes the proof of the theorem.
As in the proof of Theorem 7.1, we can prove the following result.

Theorem 7.2 Let n > 1 be an integer coprime with 6. Then
Letting a → 1 in Theorems 7.1 and 7.2, we obtain Corollary 7.3 Let n > 1 be an integer coprime with 6. Then We shall also prove the following results.

Proof It is easy to see by induction on N that
(7.7) Note that 1

[N +1] 2N
N is the well-known q-Catalan number (see [9]), a polynomial in q. Hence, the q-binomial coefficient 2N if N + 1 is coprime with 3. It is also not difficult to prove that 4N 2N is divisible by [N + 1] whenever N + 1 is coprime with 6. Therefore, putting N = n − 1 in (7.7), we can prove that the right-hand side is congruent to 0 modulo [n] 2 . Similarly, taking N = (n − 1)/2 in (7.7), we arrive at the same conclusion. This time one [n] comes from [2N + 1] and another [n] comes from 4N 2N .

Some q-Congruences from a Quartic Transformation of Gasper and Rahman
Gasper and Rahman [12] (see also [13,Ex. 3.33]) also obtained the following quartic transformation: In this section, we shall deduce two congruences from the quartic transformation (8.1).

(8.2)
Proof Replacing q by q 2 , a by q 1−n , and b by q 2−n in (8.1), we see that the left-hand side terminates at k = (n − 1)/2, while the right-hand side vanishes. (Note that we cannot make such a replacement if n ≡ 1, 3 (mod 8).) Namely, we have Since q n ≡ 1 (mod n (q)), we immediately obtain (8.2) from the above identity.
It is not difficult to see that the congruence (8.2) can also be derived from the following quartic summation formula of Gasper [11] (see also [13,Ex. 3.30]): Theorem 8.2 Let n be a positive integer with n ≡ 5, 7 (mod 8). Then mod n (q)).

(8.3)
Proof Replacing q by q 2 , a by q 1−n , and b by q 3 in (8.1), we see that the left-hand side again terminates at k = (n − 1)/2, while the right-hand side vanishes. That is, The proof of (8.3) then follows from the above identity and the fact q n ≡ 1 (mod n (q)).
We have the following two related conjectures.

Some q-Congruences from a New 12 11 Transformation
In this section, we shall deduce some q-congruences from Theorem A.1, a new 12 φ 11 transformation formula, whose proof we give in the appendix. Although all of the q-congruences are modulo n (q), the q = 1 cases sometimes can be generalized to supercongruences modulo higher powers (see Conjectures 12.6 and 12.7 in the next section).

Theorem 9.2 Let n ≡ 2 (mod 3) be an integer and n > 2. Then
Proof Replacing q → q 3 and then letting a = q −1−n and b = c = d = q −1 in (A.2), we obtain because the right-hand side of (A.2) contains the factor (q 2−n ; q 3 ) ∞ , which vanishes for n ≡ 2 (mod 3). It is easy to see that the denominator of (9.4) is relatively prime to n (q) for n > 2. Therefore, applying q n ≡ 1 (mod n (q)), we obtain the desired congruence in (9.3a). Similarly (see the proof of (9.1b) and (9.1c)), we can prove (9.3b) and (9.3c).

Some Other q-Congruences from the q-Dixon Sum
By using the q-Dixon sum [13, Eq. (II.13)], the first author and Zudilin [28,Thm. 4.12] proved the following result.
For n > 1, taking the limit as a, b, c → 1 in (10.4) we are led to (10.5).
We conjecture that the following stronger version of (10.5) is also true.

Conjecture 10.2 Let n ≡ 1 (mod 4) be an integer and n > 1. Then
Similarly to the proof of Theorem 10.1, taking q → q 4 , a → a 2 q 2 , and b = c = q −2 in (10.1), we can prove the following result.

Theorem 10.3 Let n > 1 be an odd integer. Then
in particular, Note that, for n ≡ 3 (mod 4), we can prove the following three-parametric congruence: 1 − a 2 q 2n )); (10.9) Besides, for the q = −1 case of (10.8), it seems that the corresponding congruence can be strengthened as follows.
Theorem 10.5 Let n ≡ 3 (mod 4) be a positive odd integer. Then in particular, We have the following conjectures.

Conjecture 10.7 Let n ≡ 3 (mod 4) be a positive integer. Then
). (10.11) It is easy to see that the congruence (10.11) is true modulo n (q) n (−q) by taking a, b → 1, and c → −1 in (10.2). Moreover, it is also true when q = 1 and n = p is an odd prime, since Tauraso observed that While a q-analogue of (10.12) was given by the first author [16], namely n k=0 q −k 2k k 2 (−q k+1 ; q) 4 n−k = q −n [2n + 1] 2 2n n

Some q-Congruences from a Double Series Transformation of Ismail, Rahman and Suslov
In [32, Thm. 1.1] Ismail, Rahman, and Suslov derived the following transformation formula: aq 2k )(a, b, c, d, e, f ; q) provided |a 2 q 2 /bcde f | < 1. If in (11.1) we replace q by q 3 , take a = g = q, h = aq, and b = c = d = e = f = q 2 , and suitably truncate the sum, then the following "divergent" q-supercongruence appears to be true.
On the other hand, if in (11.1) we replace q by q 3 , take a = g = q −1 , h = aq −1 , and b = c = d = e = f = q, and suitably truncate the sum, then the following "divergent" q-supercongruence appears to be true. Conjecture 11.2 Let n > 2 be a positive integer with n ≡ 2 (mod 3). Then Furthermore, the above congruence holds modulo n (q) 3 when a = 1.
If in (11.1) we replace q by q 4 , take a = b = c = d = e = f = q, g = q −1 , h = aq −1 , and suitably truncate the sum, then the following q-supercongruence appears to be true.

Conjecture 11.3 Let n be a positive integer with n ≡ 3 (mod 4). Then
Furthermore, the above congruence holds modulo [n] n (q) 3 when a = 1.
On the other hand, if in (11.1) we replace q by q 4 , take −3 , and suitably truncate the sum, then the following q-supercongruence appears to be true.
Ismail, Rahman, and Suslov [32,Eq. (5.4)] also noted the following transformation formula (which can be obtained from (11.1) by taking d = aq/c and h = 0): If in (11.2) we replace q by q 4 , take a = b = c = e = f = q −2 , g = q 5 , and truncate the sum, then the following q-supercongruence appears to be true.

Conjecture 11.5 Let n be a positive integer with n ≡ 3 (mod 8). Then
As before, we can show that all the congruences in Conjectures 11.1-11.5 are true modulo n (q). For example, we have the following parametric generalization of the congruence (11.3) modulo n (q) n (−q). Theorem 11.6 Let n be a positive integer with n ≡ 3 (mod 8). Then, modulo (11.2). Then the left-hand side terminates at k = (n + 1)/2 because of the factor (q −2−2n ; q 4 ) k in the numerator, while the right-hand side vanishes because of the factor (q 2−2n ; q 4 ) ∞ . The described specialization thus yields the following identity: Since q 2n ≡ 1 (mod n (q) n (−q)), we immediately deduce the desired congruence from the above identity.

Concluding Remarks and Further Open Problems
Most of the congruences in the manuscript [28] are modulo [n](1 − aq n )(a − q n ). However, the congruence (3.2) does not hold modulo [n](1 − aq n )(a − q n ) in general. We only have a generalization of (3.2) with a = 1.
It is easy to see that the following generalization of (2.1b) in Theorem 2.1 is true.
Letting a = 1 in Theorem 3.3, we see that the congruence (12.1) holds modulo n (q) 3 . Therefore, Theorem 12.1 is equivalent to the left-hand side of (12.1) being congruent to 0 modulo [n]. By (3.2), we see that the left-hand side of (12.1) is congruent to 0 modulo n (q). And the same technique to prove congruences modulo [n] from congruences modulo n (q) as used in the proofs of (3.7a) and (3.7b) still works here.
We conjecture that the following generalization of the second part of Theorem 2.3 is true.

Conjecture 12.2 Let n be a positive integer with n ≡ 2 (mod 3). Then
We also have the following similar conjecture.

Conjecture 12.3
Let n > 1 be a positive integer with n ≡ 1 (mod 3). Then Note that, similar to the proof of Theorem 2.3, we can show that the above congruence holds modulo [n] n (q). We point out that q-congruences modulo [n] n (q) 3 or n (q) 4 are very difficult to prove. As far as we know, the following result due to the first author and Wang [26], is the first q-congruence modulo [n] n (q) 3 in the literature that is completely proved. It is natural to ask whether there is a complete q-analogue of Long's supercongruence (1.4). Inspired by the q-congruences in the previous sections, we shall propose the following conjecture.

Conjecture 12.4 Let n be a positive odd integer. Then
Note that the left-hand side is not a truncated form of (A.2) with q → q 4 and a = b = c = d = q. Therefore, even for the case modulo n (q), the above conjecture is still open. Moreover, we cannot find any parametric generalization of the above conjecture, though one would believe that such a generalization should exist.
Similarly, the following conjecture seems to be true.

Conjecture 12.5 Let n > 1 be an odd integer. Then
For the q = 1 case of (9.1b), much more seems to be true. Numerical computations suggest the following result.

Conjecture 12.6 Let p ≡ 1 (mod 3). Then
We also have a similar conjecture related to (9.3b).

Conjecture 12.7 Let p ≡ 2 (mod 3). Then
Unfortunately, we failed to find complete q-analogues of the above two conjectures. In particular, we do not know how to use the creative microscoping method to tackle them.
In [16,Conj. 5.4] the first author has made the following conjecture. Note that the congruences (12.2a) for r = 1, 2 and (12.2b) for r = 1 have been proved by the first author [16] himself, and the congruence (12.2b) for r = 2 has been established by the first author and Wang [26].

Conjecture 12.8 Let n and r be positive integers. Then
In this section, we shall prove the following weaker form of the above conjecture. Similarly, we consider the general very-well-poised 2d φ 2d−1 series where we replace q by q d and take all upper parameters to be q −1 . Then the following generalization of Conjecture 12.3 appears to be true.
Remark Since the submission of the original version of this paper (which also appeared as a preprint on the arXiv) and the present final version, relevant developments have taken place.  [13,Ex. 8.15]). Observe that the two 4 φ 3 series on the right-hand side are not balanced, nor well-poised. However, they satisfy the remarkable property that the quotient (not the product!) of corresponding upper and lower parameters is throughout the same, namely b/q.
By replacing a, b, c, d in (A.2) by q a , q b , q c , q d , respectively, and letting q → 1 − we obtain the following transformation between a nonterminating very-well-poised 9 F 8 series into two multiples of nonterminating 4 F 3 series. (For the notion of a hypergeometric r F s series, see [2]. In the following, we employ the condensed notation for products of Pochhammer symbols, (a 1 , . . . , a m ) k = (a 1 ) k · · · (a m ) k .) where, for convergence, (b) < 3 4

. The transformation in (A.3) extends [10, Eq. (3.3)].
Proof of Theorem A. 1 We would like to take n → ∞ in (A.1) but the series on the righthand side has large terms near the end compared to those in the middle of the series which prevents us from taking the term-by-term limit directly. We thus apply a similar analysis as applied by Bailey [5,Eq. 8.5(3)] in his derivation of the nonterminating Watson transformation (who started with the terminating balanced very-well-poised 10 φ 9 transformation to derive a transformation of a nonterminating very-well-poised 8 φ 7 series into two multiples of balanced 4 φ 3 series), see also [13,Sec. 2.10]. In (A.1), we first replace n by 2m + 1. Then we write the series on the right-hand side as Notice that if in (A.2) we take d = a/c the first series on the right-hand side reduces to 1. (If instead d = ab 2 /c then the second series on the right-hand side reduces to 1. The resulting series is equivalent to (A.5) by the substitution c → c/b.) We thus have the following nonterminating very-well-poised 12 φ 11 summation: Corollary A. 2 We have If in (A.2) we take (instead of d = a/c which led to Corollary A.2) d = aq/c the prefactor of the first series on the right-hand vanishes. (If instead d = ab 2 /cq then the prefactor of the second series on the right-hand vanishes. The resulting series is equivalent to (A.7) by the substitution c → ab/c.) We thus have the following nonterminating very-well-poised 10 φ 9 transformation: Corollary A. 4 We have where |q/b| < 1.
We also record another (simpler) special case of (A.1), obtained by taking d → ∞. Alternatively, it can be obtained from Theorem A.1 by choosing d = q −n .
The continuous q-ultraspherical polynomials, which depend on a parameter β and the base q, are given by C n (x; β | q) = n k=0 (β; q) k (β; q) n−k (q; q) k (q; q) n−k e i(n−2k)θ , x = cos θ. (A.9) (Note that θ need not be real.) They were originally considered by Rogers [48] in 1884 (not aware of their orthogonality) in the pursuit of (what is now called) the Rogers-Ramanujan identities. These functions, which can be written as C n (x; β | q) = (β; q) n (q; q) n e inθ 2 φ 1 β, q −n q 1−n /β ; q, qe −2iθ β , (A. 10) are polynomials in x of degree n.
Corollary A. 7 Rogers' linearization formula for the continuous ultraspherical polynomials in (A.12) is true.
Proof In Theorem A.6 choose μ = m and ν = n for two nonnegative integers m and n. The identity (A.15) then reduces, after dividing both sides by z n+m , to (A.12).