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 Wadim 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 ${}_{12}\phi_{11}$ series. Also, the nonterminating $q$-Dixon summation formula is used. A special case of the new ${}_{12}\phi_{11}$ transformation formula is further utilized to obtain a generalization of Rogers' linearization formula for the continuous $q$-ultraspherical polynomials.


1.2)
Here and throughout the paper, p always denotes an odd prime and Γ p (x) is the p-adic Gamma function. The congruence (1.2) was later proved by McCarthy and Osburn [30] 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 [27] 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 [4], Kilbourn [24] succeeded in proving Van Hamme's (M.2) supercongruence: where a p is the p-th coefficient of a weight 4 modular form Applying Whipple's 7 F 6 transformation formula, Long [28] proved that (1.4) which in view of the supercongruence (1.3) can be written as 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 (a; q) k := (a; q) ∞ (aq k ; q) ∞ , where (a; q) ∞ = j 0 (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 r φ s a 1 , a 2 , . . . , a r b 1 , . . . , b s ; q, z := ∞ k=0 (a 1 , a 2 , . . . , a r ; q) k (q, b 1 , . . . , b s ; q) k (−1) k q ( k 2 ) 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 [ 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: = (aq, aq/de; q) ∞ (aq/d, aq/e; q) ∞ 3 φ 2 aq/bc, d, e aq/b, aq/c ; q, aq de . (1.8) Other transformations we make use of is 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, to special functions, we refer to the text books by Gasper and Rahman [13], and by Andrews, Askey and Roy [3], 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 n-th cyclotomic polynomial Φ n (q) is given by Φ n (q) := 1 k n gcd(n,k)=1 where ζ is an n-th primitive root of unity. It is clear that Φ n (q) is a polynomial in q with integer coefficients. Further, in particular, Φ p (q) = [p] for 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]. We refer the reader to [1,8,18,26,36,40,41] for some interesting q-congruences.

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 Φ n (q) 3 but not modulo [n]Φ n (q) 2 in general, we conclude that (2.1a) and (2.1b) are in fact different congruences. Of course, when n = p is an odd prime and q = 1, they are both equivalent to (1.4) modulo p 3 . The proof of Theorem 2.1 is deferred to Section 3.
Theorem 2.2. Let n be a positive odd integer. Then Note that, just like in Theorem 2.1, the two congruences (2.4a) and (2.4b) are not equivalent. The proof of Theorem 2.2 is deferred to Section 3.

(2.5)
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 We also confirm the second congruence in [20,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 Section 4. In Section 3, we shall prove Theorems 2.1 and 2.2 using the creative microscoping method developed by the first author and Zudilin [20]. Roughly speaking, to prove a q-supercongruence modulo Φ n (q) 3 , we prove its generalization with an extra parameter a so that the corresponding congruence holds modulo Φ n (q)(1 − aq n )(a − q n ). Since the polynomials Φ n (q), 1 − aq n , and a − q n are 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 [37].
Further, we introduce a new idea for proving some congruences modulo Φ n (q). In many instances in this paper, the congruences (n−1)/2 k=0 a n,k ≡ 0 (mod Φ n (q)) are proved by simply showing a k + a (n−1)/2−k ≡ 0 (mod Φ n (q)) (instead of, say, evaluating certain infinite series at roots of unity which was illustrated in [20]).
The proofs of Theorems 2.3 and 2.4 in Section 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 Sections 3 and 4, and of the further results from Section 5, are based on Watson's 8 φ 7 transformation formula. We also confirm a three-parametric q-congruence conjecture in Section 6 based on a quadratic transformation formula of Rahman. Further, in Section 7 we deduce some q-congruences from a cubic transformation formula of Gasper and Rahman. Similarly, in Section 8 we deduce some q-congruences from a quartic transformation formula of Gasper and Rahman. The q-supercongruences in Section 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 Section 10 some q-supercongruences are deduced from the q-Dixon summation. In Section 11 we deduce q-super congruences -most of them only conjecturalfrom a double series transformation of Ismail, Rahman and Suslov. Finally, in Section 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]Φ n (q) 3 for n ≡ 2 (mod 3).

Proof of Theorems 2.1 and 2.2
We first give the following lemma.
We now use the above lemma to prove the following result which was originally conjectured by the first author and Zudilin [20,Conj. 5.6].
We shall establish the following two-parameter generalization of Theorem 2.1.
Theorem 3.3. Let n be a positive odd integer. Then, modulo Φ n (q)(1 − aq n )(a − q 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 Φ n (q), the k-th and ((n−1)/2− k)-th terms on the left-hand side of (3.2) cancel each other, i.e., (aq, q/a, bq; q 2 ) k (q; q 2 ) 3 k (aq 2 , q 2 /a, q 2 /b; q 2 ) k (q 2 ; q 2 ) 3 k q b k (mod Φ n (q)).
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 Φ n (q), and therefore the congruence (3.2) also holds modulo Φ n (q). Since Φ n (q), 1 − aq n and a − q n are 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 Φ 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 Φ 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 Φ 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, For n > 1, let ζ = 1 be an n-th root of unity, not necessarily primitive. That is, ζ is a primitive root of unity of odd degree d | n. If c q (k) denotes the k-th 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 We have which means that the sums n−1 k=0 c q (k) and c q (k) are both divisible by the cyclotomic polynomial Φ d (q). Since this is true for any divisor d > 1 of n, we conclude that they are divisible by 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).

Proof of Theorems 2.3 and 2.4
We shall prove the following common generalization of Theorems 2.3 and 2.4. Then Proof. Let α and j be integers. Since and 1 − q αn ≡ 0 (mod Φ n (q)), we obtain Similarly, we have It is clear that (q d+1 ; q d ) (n−1)/d in the numerator has the factor 1 − q n and is therefore divisible by Φ n (q), while the denominator is coprime with Φ n (q). This proves (4.1) for the first case.
Similarly, if n ≡ −1 (mod d), then, modulo Φ 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 (1 − q (d−1)n )(1 − q (2−d)n ) and is therefore divisible by Φ n (q) 2 , and again the denominator is coprime with Φ n (q). This proves (4.1) for the second 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 and further (again as on the proof of Theorem 2.1) we can show that This completes the proof of (2.7).
It appears that the following generalization with one more parameter b is still true. Then

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 [20,Thm. 4.9] with r = −1, d = 2 and a = 1 gives In this section, we shall give some similar congruences.
Theorem 5.1. Let n > 1 be a positive odd integer. Then 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 righthand side of (5.3) as It is easy to see that This proves that the congruence (5.2) holds modulo (1 − aq n )(a − q n ).
On the other hand, by Lemma 3.1, for 0 k (n + 1)/2, we have It follows that the k-th and ((n + 1)/2 − k)-th terms on the left-hand side of (5.2) cancel each other modulo Φ 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 [4k − 1] = [n] is a factor of the summand. In total, this proves that the congruence (5.2) also holds modulo Φ n (q). Since the polynomials Φ n (q), 1 − aq n and a − q n are coprime with one another, the proof of (5.2) is complete.
Proof. We first establish the following congruence: Letting q → q 2 and c → 0 followed by a = b = q −1 , d = q −1−n and e = q −1+n in the limiting case of Watson's transformation formula (1.8), we obtain By the q-Chu-Vandermonde summation formula [13, Appendix (II.6)], for odd n > 1, we have Letting q → q −1 in the above equality, we see that the summation on the right-hand side of (5.7) is equal to 0. This proves that the congruence (5.6) 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 k-th and ((n + 1)/2 − k)-th terms on the left-hand side of (5.6) are congruent to 0 modulo Φ 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 Φ n (q). This completes the proof of (5.6).
Let c q (k) denote the k-th 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 The parts of the denominators in (5.10) which contain the parameter a are the factors (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 Φ n (q). On the other hand, the limit of (1 − aq n )(a − q n ) as a → 1 has the factor Φ n (q) 2 . Therefore, the limiting case a → 1 of the congruence (5.10) reduces to (5.5) modulo Φ 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 a positive odd integer. Then Proof. The proof is similar to that of Theorem 5.3. We first establish for odd n > 1. Letting q → q 2 and c → 0 followed by a = b = q, d = q −1−n and e = q −1+n in (1.8), we obtain As we have already mentioned in the proof of Theorem 5.3, the summation on the righthand 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 k-th and ((n − 1)/2 − k)-th terms on the left-hand side of (5.12) are congruent to 0 modulo Φ 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 Φ 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 Φ 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 Φ n (q) when taking the limit as a → 1. Therefore, the congruence (5.11) holds modulo Φ 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 Φ 3 (q) for n = 3. Let c q (k) denote the k-th 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).
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 end this section with 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. Similarly as in the proof of Theorem 4.1, we can confirm it for b = 1.
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)).
Here we would like to propose a supercongruence similar to (6.6).

Conjecture 6.2. We have
Unfortunately, we were not able to find any q-analogue of (6.7), 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 (6.8) and (6.9) are q-analogues of the following supercongruence due to Guillera and Zudilin [15]: The congruence (6.9) with M = n − 1 was first established by the first author [17] using the q-WZ method. The congruence (6.8) is new. Motivated by Conjecture 6.2, we would like to raise the following problem. Problem 6.3. Is there a "3k − 1 version" of the supercongruence (6.10)?
This completes the proof of the theorem.
Similarly as in the proof of Theorem 7.1, we can prove the following result.

Proof. It is easy to see by induction on N that
2N N is the well-known q-Catalan number (see [9]), a polynomial in q.
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).
Theorem 8.1. Let n be a positive integer with n ≡ 5, 7 (mod 8). Then 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 Φ n (q)), we immediately obtain (8.2) from the above identity.
The proof of (8.3) then follows from the above identity and the fact q n ≡ 1 (mod Φ n (q)).
We have the following two related conjectures.
In particular, if p ≡ 5 (mod 8), then Conjecture 8.4. The congruence (8.3) is still true modulo Φ n (q) 3 . In particular, if p ≡ 5, 7 (mod 8), then 9. Some q-congruences from a new 12 φ 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 qcongruences are modulo Φ 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 9.1. Let n ≡ 1 (mod 3) be a positive integer and n > 1. Then and Proof. Replacing q → q 3 and then letting a = q 1−n , b = c = d = q in (A.2), we obtain because the right-hand side of (A.2) contains the factor (q 1−n ; q 3 ) ∞ , which vanishes for n ≡ 1 (mod 3). Since q n ≡ 1 (mod Φ n (q)), we immediately deduce (9.1a) from (9.2) . Similarly, if we change c = q to c → 0 in the above procedure, then we can prove (9.1b), while if we change c = d = q to c, d → 0 then we are led to (9.1c).
Theorem 9.2. Let n ≡ 2 (mod 3) be a positive integer and n > 2. Then Proof. Replacing q → q 3 and then letting 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 Φ n (q) for n > 2. Therefore, applying q n ≡ 1 (mod Φ 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).
We conjecture that the following stronger version of (10.5) is also 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.
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]Φ n (q) 3 when a = 1.
On the other hand, if in (11.1) we replace q by q 4 , take a = b = c = d = e = f = q −1 , g = q −3 , h = aq −3 , and suitably truncate the sum, then the following q-supercongruence appears to be true.
Ismail, Rahman and Suslov [23,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 Similarly as before, we can show that all the congruences in Conjectures 11.1-11.5 are true modulo Φ n (q). For example, we have the following parametric generalization of the congruence (11.3) modulo Φ n (q)Φ n (−q).

Concluding remarks and further open problems
Most of the congruences in the manuscript [20] 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.
In particular, we conjecture that the following generalization of (2.1b) in Theorem 2.1 is true.
Conjecture 12.1. Let n be a positive odd integer. Then, modulo [n]Φ n (q) 2 , Letting a = 1 in Theorem 3.3, we see that the congruence (12.1) holds modulo Φ n (q) 3 . Therefore, Conjecture 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 Φ n (q). But the same technique to prove congruences modulo [n] from congruences modulo Φ n (q) as used in the proofs of (3.7a) and (3.7b) does not work here, because the argument of the series is q/b and not just a power of q.
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]Φ n (q). We point out that q-congruences modulo [n]Φ n (q) 3 or Φ n (q) 4 are very difficult to prove. As far as we know, the following result due to the first author and Wang [19], is the unique q-congruence modulo [n]Φ n (q) 3 in the literature that is completely proved. (Several similar conjectural q-congruences are collected in [20].) 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 section, 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 Φ n (q), the above conjecture is still open. Moreover, we cannot find any parametric generalization of the above conjecture, although one would believe that such a generalization should exist.
Similarly, the following conjecture seems to be true.
Conjecture 12.5. Let n > 1 be a positive odd integer. Then For the q = 1 case of (9.1b), much more seems to be true. Numerical computations suggest the following result.
Conjecture 12.7. 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.
Conjecture 12.8. Let n and r be positive integers. Then 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 [19]. Furthermore, the congruences (12.2a) and (12.2b) for q = 1 have recently been confirmed by Ni and Pan [32].
In this section, we shall prove the following weaker form of the above conjecture.
Finally, we consider the general very-well-poised 2d φ 2d−1 series (which satisfies Slater's transformation [13, r = d and b 2r = a in Eq. (5.5.2)]) where we replace q by q d and take all upper parameters to be q. Then the following further generalization of Conjecture 12.2 appears to be true.
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.
This results extends Gasper's [10, Eq. (3. 2)] (see also [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 [3]. 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, ℜ(b) < Proof of Theorem A.1. We would like to take n → ∞ in (A.1) but the series on the right-hand 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 [6,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 after which in each of the sums we can term-wise let m → ∞ (which is justified by Tannery's theorem [39, p. 292] under the restriction |q/b| < 1). The identity in (A.2) thus follows.
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.5. We have n k=0 1 − aq 2k 1 − a (a, b, c, ab/c, abq n , q −n ; q) k (q, aq/b, aq/c, cq/b, q 1−n /b, aq n+1 ; q) k (aq/b; q) 2k (ab; q) 2k q b k = (aq, ab/c; q) n (ab, aq/c; q) n n k=0 (b, c, q −n c/a, q −n ; q) k (q, cq/b, cq 1−n /ab, q 1−n /b; q) k q b 2k . (A.8) We use Corollary A.5 to provide a generalization of Rogers' linearization formula for the continuous q-ultraspherical polynomials in (A.12).