Abstract
We give a q-congruence whose specializations \(q=-1\) and \(q=1\) correspond to supercongruences (B.2) and (H.2) on Van Hamme’s list (in: p-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Applied Mathematics, vol 192. Dekker, New York, pp 223–236, 1997):
where \(p>2\) is prime,
and a(p) is the pth coefficient of the modular form \(q\prod _{j=1}^\infty (1-q^{4j})^6\) (of weight 3). We complement our result with a general common q-congruence for related hypergeometric sums.
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1 Introduction
The formula of Bauer [1] from 1859,
is one of traditional targets for different methods of proofs of hypergeometric identities. Its special status is probably linked to the fact that it belongs to a family of series for \(1/\pi \) of Ramanujan type, after Ramanujan [21] brought to life in 1914 a long list of similar looking equalities for the constant but with a faster convergence. Identity (1.1) is a particular instance of \(_4F_3\) hypergeometric summation (known to Ramanujan) but there are several proofs of it, including the original one [1] of Bauer, that do not require any knowledge of hypergeometric functions. One notable—computer—proof of (1.1) was given in 1994 by Ekhad and Zeilberger [2] using the Wilf–Zeilberger (WZ) method of creative telescoping.
It was observed in 1997 by Van Hamme [28] that many Ramanujan’s and Ramanujan-like evaluations have nice p-adic analogues; for example, the congruence
(tagged (B.2) on Van Hamme’s list) is valid for any prime \(p>2\) and corresponds to the equality (1.1). The congruence (1.2) was first proved by Mortenson [19] using a \(_6F_5\) hypergeometric transformation; it later received another proof by one of these authors [29] via the WZ method [in fact, using the very same ‘WZ certificate’ as in [2] for (1.1)]. Notice that (1.2) is an example of supercongruence meaning that it holds modulo a power of p greater than 1.
Another entry on Van Hamme’s 1997 list [28], tagged (H.2), is the congruence
again for any \(p>2\) prime, and \(\Gamma _p(x)\) is the p-adic Gamma function. Van Hamme not only observed but also proved (1.3) in [28], and it was later generalized by Sun [23, 24, Theorem 2.5], Guo and Zeng [12, Corollary 1.2], Long and Ramakrishna [17], Liu [15, 16, Theorem 1.5] in different ways. For example, Long and Ramakrishna [17, Theorem 3] gave the following generalization of (1.3):
Recently, these authors [14, Theorem 2] proved that, for any positive odd integer n, modulo \(\Phi _n(q)^2\),
Here and in what follows, \(\Phi _n(q)\) denotes the nth cyclotomic polynomial; the q-shifted factorial is given by \((a;q)_0=1\) and \((a;q)_n=(1-a)(1-aq)\ldots (1-aq^{n-1})\) for \(n\geqslant 1\) or \(n=\infty \), while \([n]=[n]_q=1+q+\cdots +q^{n-1}\) stands for the q-integer. Van Hamme [27, Theorem 3] also proved that
in view of \(\Gamma _p(1/2)^2=-1\) for \(p\equiv 1\;({\text {mod}}4)\), by letting \(q\rightarrow 1\) in (1.5) for \(n=p\) we immediately obtain (1.3).
One feature of (1.3) (not highlighted in [28]) is its connection with the coefficients
of CM modular form \(q\prod _{j=1}^\infty (1-q^{4j})^6\) of weight 3, namely, the congruence
This served as a main motivation in [14] for not only establishing (1.5) but also speculating on possible q-deformation of modular forms.
For some other recent progress on q-analogues of supercongruences, the reader is referred to [4, 5, 7,8,9,10,11, 13, 20, 22, 26, 29]. In particular, the authors [13] introduced and executed a new method of creative microscoping to prove (and reprove) many q-analogues of classical supercongruences and also raised some problems on q-congruences. Using this method, the first author [6] gave a refinement of (1.5) modulo \(\Phi _n(q)^3\) for \(n\equiv 3\;({\text {mod}}4)\), in other words, a q-analogue of (1.4) for \(p\equiv 3\;({\text {mod}}4)\).
A goal of this note is to present the following new q-analogue of Van Hamme’s supercongruence (1.3).
Theorem 1.1
Let n be a positive odd integer. Then
Note that \(\Phi _n(q)\Phi _n(-q)=\Phi _n(q^2)\) for odd indices n.
The \(n\equiv 3\;({\text {mod}}4)\) case of Theorem 1.1 confirms a conjecture of these authors [13, Conjecture 4.13], which states that, for \(n\equiv 3\;({\text {mod}}4)\),
It is not difficult to verify that
for \(p\equiv 3\;({\text {mod}}4)\), where \((a)_n=a(a+1)\ldots (a+n-1)\) denotes the rising factorial (also known as Pochhammer’s symbol). Therefore, the q-congruence (1.7) reduces to (1.4) for \(p\equiv 3\;({\text {mod}}4)\) when \(n=p\) and \(q\rightarrow 1\), and it reduces to (1.3) for \(p\equiv 1\;({\text {mod}}4)\) when \(n=p\) and \(q\rightarrow 1\). Moreover, letting \(n=p\) and \(q\rightarrow -1\) in (1.7), we immediately get (1.2). Thus, Theorem 1.1 presents a common q-analogue of supercongruences (1.2) and (1.3). We point out that other different q-analogues of (1.2) have been given in [7, 8].
Recently, Mao and Pan [18] (see also Sun [25, Theorem 1.3]) proved that, if \(p\equiv 1\;({\text {mod}}4)\) is a prime, then
In this note, we prove the following q-analogue of (1.8).
Theorem 1.2
Let \(n>1\) be an odd integer. Then
The \(n\equiv 1\;({\text {mod}}4)\) case of Theorem 1.2 also confirms a conjecture of the first author and Schlosser [11, Conjecture 10.2].
For n prime, letting \(q\rightarrow 1\) in Theorem 1.2 we obtain the following generalization of (1.8).
Corollary 1.3
Let p be an odd prime. Then
On the other hand, for n prime and \(q\rightarrow -1\) in Theorem 1.2, we are led to the following result:
It should be mentioned that a different q-analogue of (1.9) was given in [13, Theorem 4.9] with \(r=-1\), \(d=2\) and \(a=1\) (see also [11, Section 5]).
Moreover, for the summation formula
we have the following q-analogue.
Theorem 1.4
We have
Both Theorems 1.1 and 1.2 are particular cases of a more general result, which we state and prove in the next section, while Theorem 1.4 follows from a classical q-identity.
2 A Family of q-Congruences from the q-Dixon Sum
In this section we establish the following family of one-parameter q-congruences.
Theorem 2.1
Let \(n\geqslant 1\) be an odd integer and \(\ell \) an integer with \(0\leqslant \ell \leqslant (n-1)/2\). Then
Note that the q-congruence (2.1) remains true when the sum is over k from 0 to \((n-1)/2+\ell \), since \((q^{2-4\ell };q^4)_k/(q^4;q^4)_k\equiv 0\;({\text {mod}}\Phi _n(q^2))\) for \((n-1)/2+\ell <k\leqslant n-1\). Furthermore, when \(\ell =0\) and \(\ell =1\) (hence \(n\ge 3\)) the theorem reduces to Theorems 1.1 and 1.2, respectively.
The following easily proved q-congruence (see [11, Lemma 3.1]) is necessary in our derivation of Theorem 2.1.
Lemma 2.2
Let n be a positive odd integer. Then, for \(0\leqslant k\leqslant (n-1)/2\), we have
Like the proofs given in [13], we start with the following generalization of (1.7) with an extra parameter a.
Theorem 2.3
Let \(n>1\) be an odd integer and \(0\leqslant \ell \leqslant (n-1)/2\). Then
Proof
Performing the parameter substitutions \(q\mapsto q^4\), \(a\mapsto q^{2-4\ell }\), \(b\mapsto bq^{2-4\ell }\) and \(c\mapsto cq^{2-4\ell }\) in the q-Dixon sum [3, Appendix (II.13)], we obtain
Since n is odd, putting \(b=q^{-2n}\) and \(c=q^{2n}\) in (2.3) we see that the left-hand side terminates and is equal to
while the right-hand side becomes
This proves that the q-congruence (2.2) holds modulo \(1-aq^{2n}\) or \(a-q^{2n}\).
On the other hand, by Lemma 2.2, for \(0\leqslant k\leqslant (n-1)/2+\ell \), modulo \(\Phi _n(q)\) we have
where we used \(q^n\equiv 1\;({\text {mod}}\Phi _n(q))\) in the last step. Using the above q-congruence we can easily check that, for odd \(n>1\) and \(0\leqslant k\leqslant (n-1)/2+\ell \), sum of the kth and \(((n-1)/2+\ell -k)\)th summands on the left-hand side of (2.2) is congruent to 0 modulo \(\Phi _n(-q)\) (or modulo \(\Phi _n(q^2)\) if \(n\equiv 3-2\ell \;({\text {mod}}4)\)). It follows that
Clearly, the right-hand side of (2.1) is congruent to 0 modulo \(\Phi _n(-q)\) if \(n+2\ell \equiv 1\;({\text {mod}}4)\) and modulo \(\Phi _n(q^2)\) if \(n+2\ell \equiv 3\;({\text {mod}}4)\). Therefore, the q-congruence (2.2) holds modulo \(\Phi _n(-q)\) if \(n+2\ell \equiv 1\;({\text {mod}}4)\) and modulo \(\Phi _n(q^2)\) if \(n+2\ell \equiv 3\;({\text {mod}}4)\). Since the polynomials \(1-aq^{2n}\), \(a-q^{2n}\) and \(\Phi _n(-q)\) (or \(\Phi _n(q^2)\)) are pairwise coprime, we complete the proof of (2.2). \(\square \)
Proof of Theorem 2.1
We assume that \(n>1\), since the \(n=1\) case (making \(\ell =0\) only possible) is trivial. The limits of the denominators on both sides of (2.2) as \(a\rightarrow 1\) are relatively prime to \(\Phi _n(q^2)\), since k is in the range \(0\leqslant k\leqslant (n-1)/2+\ell \). On the other hand, the limit of \((1-aq^{2n})(a-q^{2n})\) as \(a\rightarrow 1\) contains the factor \(\Phi _n(q^2)^2\). \(\square \)
Proof of Theorem 1.4
Take \(b=c=\ell =1\) in Eq. (2.3). \(\square \)
3 Discussion
The method of creative microscoping used in our proofs indicates the origin of q-congruences from infiniteq-hypergeometric identities; for example, the q-congruence (1.7) corresponds to the identity
which is just a particular instance of (2.3). Note that the limiting cases as \(q\rightarrow -1\) and \(q\rightarrow 1\) of (3.1) give the formulas (1.1) and
where
is the CM modular form from the introduction and L(f, s) denotes its L-function. This means that the q-identity (3.1) presents a common q-extension of evaluations (1.1) and (3.2)—the fact that makes it less surprising that the q-congruence (1.7) simultaneously extends (1.2) and (1.3).
The intermediate use of parametricq-hypergeometric identities in our proof of Theorem 2.1 based on the q-Dixon sum suggests that different q-congruences underlying (3.1) are possible. This is indeed the case when we analyze the formula (3.1) as the \(a=1\) specialization of
which originates from a q-analogue of Watson’s \(_3F_2\) sum [3, Appendix (II.16)]. When we choose \(a=q^n\) (or \(a=q^{-n}\)) in (3.3), for \(n>1\) odd, we get the sum terminating after \((n-1)/2\) terms on the left-hand side of (3.3), while the right-hand side vanishes if n is of the form \(4m+3\) and it becomes equal to
if \(n=4m+1\). This means that modulo \((a-q^n)(1-aq^n)\) we have
for any \(N\ge (n-1)/2\). The limiting \(a\rightarrow 1\) case of the congruences can be shown to be
modulo \(\Phi _n(q)^2\Phi _n(-q)\). This is quite similar in spirit to (1.5), though still far from constructing q-analogues for the coefficients a(p) in (1.6) of the modular form \(f(\tau )\). The latter means that a hunt for q-rational functions, which equal the left-hand side of (1.5) or (3.4) modulo \(\Phi _n(q)^2\) and specialize to a(n) as \(q\rightarrow 1\) (at least for n prime), is still on its way. Such q-rational functions are also expected to be self-reciprocal, that is, invariant under the involution \(q\mapsto 1/q\), as all the left- and right-hand sides in (1.5), (1.7), (3.4) and also (2.1) are.
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Victor J. W. Guo was supported by the National Natural Science Foundation of China (grant 11771175). Wadim Zudilin was supported by JSPS Invitational Fellowships for Research in Japan (fellowship S19126).
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Guo, V.J.W., Zudilin, W. A Common q-Analogue of Two Supercongruences. Results Math 75, 46 (2020). https://doi.org/10.1007/s00025-020-1168-7
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DOI: https://doi.org/10.1007/s00025-020-1168-7