Abstract
In this paper, a couple of q-supercongruences for truncated basic hypergeometric series are proved, most of them modulo the cube of a cyclotomic polynomial. One of these results is a new q-analogue of the (E.2) supercongruence by Van Hamme, another one is a new q-analogue of a supercongruence by Swisher, while the other results are closely related q-supercongruences. The proofs make use of special cases of a very-well-poised \({}_6\phi _5\) summation. In addition, the proofs utilize the method of creative microscoping (which is a method recently introduced by the first author in collaboration with Wadim Zudilin), and the Chinese remainder theorem for coprime polynomials.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In 1997, Van Hamme [24] presented 13 remarkable supercongruences corresponding to Ramanujan’s or to Ramanujan-like formulas for \(1/\pi \). For instance, the two infinite series expansions
correspond to the following two supercongruences for truncated hypergeometric series:
where p is an odd prime, and \((a)_n=a(a+1)\cdots (a+n-1)\) denotes the Pochhammer symbol. The supercongruence (1.1) was first proved by Mortenson [18] using a technical evaluation of gamma functions, and later reproved by Zudilin [28] and Long [16]. Swisher [23] employed Long’s method to prove four supercongruences of Van Hamme, including (1.2) (i.e., the (E.2) supercongruence in [24]). He [11] also gave a generalization of (1.2). In 2016, Osburn and Zudilin [21] confirmed the last supercongruence conjecture of Van Hamme.
During the past few years, q-analogues of supercongruences have been investigated by many authors (see, for example, [3,4,5,6,7,8,9,10, 12,13,14,15, 19, 20, 22, 25,26,27, 29]). In particular, the first author [3, 4] gave q-analogues of (1.1) and (1.2) as follows: for any odd integer n,
and for any positive integer n with \(n\equiv 1\pmod {3}\),
Here and in what follows, \((a;q)_n=(1-a)(1-aq)\cdots (1-aq^{n-1})\) is the q-shifted factorial, \([n]=[n]_q=(1-q^n)/(1-q)\) is the q-integer, and \(\Phi _n(q)\) denotes the n-th cyclotomic polynomial in q, i.e.,
where \(\zeta \) is an n-th primitive root of unity. The first author and Zudilin [10, Theorem 3.5 with \(r=1\)] also gave another q-analogue of (1.2): for any positive integer n with \(n\equiv 1\pmod {6}\),
One of the aims of this paper is to establish the following new q-analogue of (1.2).
Theorem 1.1
Let \(n\equiv 1\pmod {6}\) be a positive integer. Then
where \(M=(n-1)/3\) or \(M=n-1\).
We shall also give the following similar result.
Theorem 1.2
Let \(n\equiv 1\pmod {3}\) be a positive integer. Then
where \(M=(n-1)/3\) or \(M=n-1\).
Note that the supercongruence (1.5) does not hold modulo \([n]\Phi _n(q)^2\) in general, even for \(n\equiv 1\pmod {6}\). We take this opportunity to point out that Theorems 1 and 2 in [8] only hold modulo \(\Phi _n(q)^3\) and \(\Phi _n(q)^2\), respectively, but do not hold modulo [n], since Lemma 3 in [8] is not true (it only holds for even integers d).
Swisher [23] also proved that, for any prime \(p\equiv 2\pmod {3}\),
A q-analogue of (1.6) was given by the first author [4, Theorem 1.5 with \((d,r)=(3,1)\)]: for any positive integer \(n\equiv 2\pmod {3}\),
In this paper, we shall give a new q-analogue of (1.6).
Theorem 1.3
Let \(n\equiv 5\pmod {6}\) be a positive integer. Then
where \(M=(2n-1)/3\) or \(M=n-1\).
Similarly, we have the following result.
Theorem 1.4
Let \(n\equiv 2\pmod {3}\) be a positive integer. Then
where \(M=(2n-1)/3\) or \(M=n-1\).
Note that the q-supercongruence (1.8) does not hold modulo \(\Phi _n(q)^3\) for \(n>2\). We shall prove Theorems 1.1, 1.2, and 1.3 modulo \(\Phi _n(q)^3\) and Theorem 1.4 by using a summation for a very-well-poised \({}_6\phi _5\) series and the ‘creative microscoping’ method introduced by the first author in collaboration with Zudilin [9]. The proof of Theorems 1.1 and 1.3 also requires the use of a lemma previously given by the present authors.
From Theorems 1.2 and 1.4, we can deduce the following supercongruences.
Corollary 1.5
Let \(p\equiv 1\pmod {3}\) be a prime. Then
where \(\Gamma _p(x)\) denotes the p-adic Gamma function.
Corollary 1.6
Let \(p\equiv 2\pmod {3}\) be an odd prime. Then
2 Proof of Theorem 1.1
We first give the following result, which is due to the present authors [6, Lemma 2.1].
Lemma 2.1
Let d, m and n be positive integers with \(m\leqslant n-1\). Let r be an integer satisfying \(dm\equiv -r\pmod {n}\). Then, for \(0\leqslant k\leqslant m\) and any indeterminate a, we have
If \(\gcd (d,n)=1\), then the above q-congruence also holds for \(a=1\).
We also need the following result to prove the truth of (1.4) modulo [n].
Lemma 2.2
Let n be a positive integer coprime with 6, and let a be an indeterminate. Then
where \(m=(n-1)/3\) if \(n\equiv 1\pmod {6}\), and \(m=(2n-1)/3\) if \(n\equiv 5\pmod {6}\).
Proof
Clearly, Lemma 2.2 is true for \(n=1\). We now assume that \(n>1\). By Lemma 2.1, we can easily deduce that the k-th and \((m-k)\)-th terms on the left-hand side of (2.1) cancel each other modulo \(\Phi _n(q)\), i.e.,
Thus, we have proved that the q-congruence (2.1) holds modulo \(\Phi _n(q)\). Since the numerator contains the factor \((q;q^3)_k\), it is easy to see that the k-th summand in (2.2) is congruent to 0 modulo \(\Phi _n(q)\) for \(m< k\leqslant n-1\). This proves the q-congruence (2.2) modulo \(\Phi _n(q)\).
Now we can prove (2.1) and (2.2) modulo [n]. Let \(\zeta \ne 1\) be an n-th root of unity, not necessarily primitive. In other words, \(\zeta \) is a primitive root of unity of degree s satisfying \(s\mid n\) and \(s>1\). Let \(c_q(k)\) stand for the k-th term on the left-hand side of (2.2), i.e.,
Taking \(n=s\) in the q-congruences (2.1) and (2.2) modulo \(\Phi _n(q)\), we get
where \(s_1=(s-1)/3\) if \(s\equiv 1\pmod {6}\), and \(s_1=(2s-1)/3\) if \(s\equiv 5\pmod {6}\). It is not difficult to see that
Therefore,
and
This proves that both of the sums \(\sum _{k=0}^{n-1}c_q(k)\) and \(\sum _{k=0}^{m}c_q(k)\) are divisible by \(\Phi _s(q)\) for any divisor \(s>1\) of n. Since
we complete the proof of (2.1) and (2.2). \(\square \)
Like most of the q-supercongruences in [9], we have the following parametric generalization of Theorem 1.1.
Theorem 2.3
Let \(n\equiv 1\pmod {6}\) be a positive integer. Then, modulo \([n](1-aq^n)(a-q^n)\),
where \(M=(n-1)/3\) or \(M=n-1\).
Proof
We start with the following summation for a very-well-poised \({}_6\phi _5\) series (see [2, Appendix (II.20)]):
(The infinite series in (2.5) converges for \(|q|<1\) and \(|aq/bcd|<1\).) Specializing (2.5) by letting \(q\mapsto q^3\), \(a=q\), \(b=q^{1-n}\), \(c=q^{1+n}\), and \(d=-q^2\), we have
This shows that both sides of (2.4) are equal for \(a=q^{-n}\) and \(a=q^n\). This means that the congruence (2.4) holds modulo \(1-aq^n\) and \(a-q^n\).
Moreover, by Lemma 2.2, the left-hand side of (2.4) is congruent to 0 modulo [n]. Since \(1-q^n\) (n is odd) is relatively prime to \(1+q^k\), we see that the right-hand side of (2.4) is also congruent to 0 modulo [n]. Noticing that \(1-aq^n\), \(a-q^n\), and [n] are pairwise coprime polynomials in q, we finish the proof of the theorem. \(\square \)
Proof of Theorem 1.1
Since \((1-q^n)^2\) contains the factor \(\Phi _n(q)^2\) and \((q^3;q^3)_M\) is coprime with \(\Phi _n(q)\), letting \(a=1\) in (2.4), we conclude that (1.4) is true modulo \(\Phi _n(q)^3\). Note that Lemma 2.2 also holds for \(a=1\). Namely, the q-congruence (1.4) is true modulo [n] and is therefore also true modulo \([n]\Phi _n(q)^2\). This completes the proof. \(\square \)
3 Proof of Theorem 1.2
We first give the following parametric generalization of Theorem 1.2: for \(n\equiv 1\pmod {3}\), modulo \(\Phi _n(q)(1-aq^n)(a-q^n)\),
where \(M=(n-1)/3\) or \(M=n-1\). The proof of (3.1) is analogous to that of (2.4). This time, we make the substitutions \(q\mapsto q^3\), \(a=q\), \(b=q^{1-n}\), \(c=q^{1+n}\), and \(d=q^2\) in (2.5) to obtain
Thus, the two sides of (3.1) are equal for \(a=q^{-n}\) and \(a=q^n\). This means that the congruence (3.1) is true modulo \(1-aq^n\) and \(a-q^n\).
Moreover, by Lemma 2.1, for \(m=(n-1)/3\) we can deduce that the k-th and \((m-k)\)-th terms on the left-hand side of (3.1) cancel each other modulo \(\Phi _n(q)\), i.e.,
(Note that we have utilized the fact that \(q^{n/2}\equiv -1\pmod {\Phi _n(q)}\) for even n.) This proves (3.1) modulo \(\Phi _n(q)\).
Finally, letting \(a=1\) in (3.1), we arrive at the q-supercongruence (1.5).
4 Proof of Theorems 1.3 and 1.4
Proof of Theorem 1.3
We first give a parametric generalization of Theorem 1.4: for \(n\equiv 5\pmod {6}\), modulo \([n](1-aq^{2n})(a-q^{2n})\),
where \(M=(2n-1)/3\) or \(n-1\). The proof of (4.1) is very similar to that of (2.4). Specializing (2.5) by \(q\mapsto q^3\), \(a=q\), \(b=q^{1-2n}\), \(c=q^{1+2n}\), and \(d=-q^2\), we have
This proves the congruence (4.1) modulo \(1-aq^{2n}\) and \(a-q^{2n}\). Moreover, the proof of (4.1) modulo [n] follows from Lemma 2.2. \(\square \)
Finally, taking \(a=1\) in (4.1), we arrive at the desired q-supercongruence (1.7).
Proof of Theorem 1.4
We have the following congruence with a parameter a: for \(n\equiv 5\pmod {6}\), modulo \((1-aq^{2n})(a-q^{2n})\),
where \(M=(2n-1)/3\) or \(M=n-1\). The congruence (4.2) is equivalent to say that both sides are equal for \(a=q^{2n}\) and \(a=q^{-2n}\). But this again follows from (2.5) by performing the parameter substitutions \(q\mapsto q^3\), \(a=q\), \(b=q^{1-2n}\), \(c=q^{1+2n}\), and \(d=q^2\). At last, letting \(a=1\) in (4.2), we get (1.8). \(\square \)
5 Proof of Corollaries 1.5 and 1.6
Proof of Corollary 1.5
Letting \(n=p\), where p is a prime congruent to \(1\pmod {3}\), and \(q\rightarrow 1\) in (1.5), we obtain
Recall that the p-adic Gamma function has the properties: for any p-adic integer x,
where \(a_0(x)\in \{1,2,\ldots ,p\}\) satisfies \(a_0(x)\equiv x\pmod {p}\). Let \(\Gamma (x)\) be the classical Gamma function. Then
By [17, Theorem 14]), for \(p\geqslant 5\), we have
and so \(\Gamma _p(\frac{p+2}{3})\Gamma _p(\frac{2-p}{3})\equiv \Gamma _p(\frac{2}{3})^2 \pmod {p^2}\). The proof then follows from the fact \(\Gamma _p(1)=(-1)^{(2p+1)/3}=-1\). \(\square \)
Proof of Corollary 1.6
Letting \(n=p\), where p is an odd prime congruent to \(2\pmod {3}\), and \(q\rightarrow 1\) in (1.8), we obtain
Further,
and by (5.1), \(\Gamma _p(\frac{2p+2}{3})\Gamma _p(\frac{2-2p}{3})\equiv \Gamma _p(\frac{2}{3})^2 \pmod {p^2}\). \(\square \)
6 Some Open Problems
Although the q-supercongruence (1.5) is not true modulo [n] in general, using the same arguments as in the proof of Theorem 1.1, we can show that, for \(n\equiv 1\pmod {3}\) and \(n>1\),
where \(M=(n-1)/3\) or \(M=n-1\). Letting \(n=p^r\) and \(q\rightarrow 1\) in the above q-congruence, we obtain the following result: for any prime \(p\equiv 1\pmod {3}\) and integer \(r\geqslant 1\),
where \(d=1,3\). Inspired by Dwork’s work [1] and Swisher’s conjectures [23, (A.3)–(L.3)], we propose the following conjecture on Dwork-type supercongruences, which is a uniform generalization of (1.9) and (6.2).
Conjecture 6.1
Let \(p\equiv 1\pmod {3}\) be a prime and let \(r\geqslant 1\). Then
where \(d=1,3\).
Note that the following Dwork-type supercongruence (see [23, (E.3)] and [4, Conjecture 5.3]) has been proved by the first author and Zudilin [9, Theorem 3.5] by establishing its q-analogue:
For any prime \(p\equiv 1\pmod {3}\) and integer \(r\geqslant 1\),
where \(d=1,3\).
We believe that the following new q-analogue of (6.3), which is also a generalization of Theorem 1.1, should be true.
Conjecture 6.2
Let \(n>1\) be an integer with \(n\equiv 1\pmod {6}\) and let \(r\geqslant 1\). Then, modulo \([n^r]\prod _{j=1}^r\Phi _{n^j}(q)^2\),
where \(d=1,3\).
Likewise, we conjecture a Dwork-type generalization of Theorem 1.2 as follows.
Conjecture 6.3
Let \(n>1\) be an integer with \(n\equiv 1\pmod {3}\) and let \(r\geqslant 1\). Then, modulo \(\prod _{j=1}^r\Phi _{n^j}(q)^3\),
where \(d=1,3\).
Data Availibility
Data sharing not applicable to this article.
References
Dwork, B.: p-adic cycles. Publ. Math. Inst. Hautes Études Sci. 37, 27–115 (1969)
Gasper, G., Rahman, M.: Basic Hypergeometric Series, vol. 96, 2nd edn. Cambridge University Press, Cambridge (2004)
Guo, V.J.W.: A q-analogue of a Ramanujan-type supercongruence involving central binomial coefficients. J. Math. Anal. Appl. 458, 590–600 (2018)
Guo, V.J.W.: q-Analogues of the (E.2) and (F.2) supercongruences of Van Hamme. Ramanujan J. 49, 531–544 (2019)
Guo, V.J.W.: A new extension of the (A.2) supercongruence of Van Hamme. Results Math. 77, 96 (2022)
Guo, V.J.W., Schlosser, M.J.: A new family of \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. Results Math. 75, 155 (2020)
Guo, V.J.W., Schlosser, M.J.: Some \(q\)-supercongruences from transformation formulas for basic hypergeometric series. Constr. Approx. 53, 155–200 (2021)
Guo, V.J.W., Schlosser, M.J.: Some q-supercongruences modulo the square and cube of a cyclotomic polynomial. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 115, 132 (2021)
Guo, V.J.W., Zudilin, W.: A \(q\)-microscope for supercongruences. Adv. Math. 346, 329–358 (2019)
Guo, V.J.W., Zudilin, W.: Dwork-type supercongruences through a creative \(q\)-microscope. J. Combin. Theory Ser. A 178, 105362 (2021)
He, B.: Some congruences on truncated hypergeometric series. Proc. Am. Math. Soc. 143, 5173–5180 (2015)
Li, L., Wang, S.-D.: Proof of a \(q\)-supercongruence conjectured by Guo and Schlosser. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 114, 190 (2020)
Liu, J.-C.: On a congruence involving \(q\)-Catalan numbers. C. R. Math. Acad. Sci. Paris 358, 211–215 (2020)
Liu, J.-C., Petrov, F.: Congruences on sums of \(q\)-binomial coefficients. Adv. Appl. Math. 116, 102003 (2020)
Liu, Y., Wang, X.: \(q\)-Analogues of two Ramanujan-type supercongruences. J. Math. Anal. Appl. 502, 125238 (2021)
Long, L.: Hypergeometric evaluation identities and supercongruences. Pacific J. Math. 249, 405–418 (2011)
Long, L., Ramakrishna, R.: Some supercongruences occurring in truncated hypergeometric series. Adv. Math. 290, 773–808 (2016)
Mortenson, E.: A \(p\)-adic supercongruence conjecture of van Hamme. Proc. Am. Math. Soc. 136, 4321–4328 (2008)
Ni, H.-X.: A \(q\)-Dwork-type generalization of Rodriguez-Villegas’ supercongruences. Rocky Mt. J. Math. 51, 2179–2184 (2021)
Ni, H.-X., Pan, H.: Some symmetric \(q\)-congruences modulo the square of a cyclotomic polynomial. J. Math. Anal. Appl. 481, 123372 (2020)
Osburn, R., Zudilin, W.: On the (K.2) supercongruence of Van Hamme. J. Math. Anal. Appl. 433, 706–711 (2016)
Straub, A.: Supercongruences for polynomial analogs of the Apéry numbers. Proc. Am. Math. Soc. 147, 1023–1036 (2019)
Swisher, H.: On the supercongruence conjectures of van Hamme. Res. Math. Sci. 2, 18 (2015)
Van Hamme, L.: Some conjectures concerning partial sums of generalized hypergeometric series. In: \(p\)-Adic Functional Analysis (Nijmegen, 1996), Lecture Notes in Pure and Appl. Math. vol. 192, Dekker, New York, pp. 223–236 (1997)
Xu, C., Wang, X.: Proofs of Guo and Schlosser’s two conjectures. Period. Math. Hungar. 85, 472–480 (2022)
Wang, X., Xu, C.: \(q\)-Supercongruences on triple and quadruple sums. Results Math. 78, 27 (2023)
Wei, C.: Some \(q\)-supercongruences modulo the fourth power of a cyclotomic polynomial. J. Combin. Theory Ser. A 182, 105469 (2021)
Zudilin, W.: Ramanujan-type supercongruences. J. Number Theory 129, 1848–1857 (2009)
Zudilin, W.: Congruences for \(q\)-binomial coefficients. Ann. Combin. 23, 1123–1135 (2019)
Acknowledgements
The authors thank the anonymous referee for a careful reading of a previous version of this paper.
Funding
Open access funding provided by Austrian Science Fund (FWF). The second author is partially supported by FWF Austrian Science Fund (Grant P 32305)
Author information
Authors and Affiliations
Contributions
All authors contributed to the preparation, writing, reviewing and editing of the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Guo, V.J. ., Schlosser, M.J. New q-Analogues of Van Hamme’s (E.2) Supercongruence and of a Supercongruence by Swisher. Results Math 78, 105 (2023). https://doi.org/10.1007/s00025-023-01885-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-01885-8
Keywords
- Basic hypergeometric series
- supercongruences
- q-congruences
- cyclotomic polynomial
- \({}_6\phi _5\) summation