q-Analogues of two Ramanujan-type supercongruences

Abstract

In this paper, we establish a new q-congruence with parameters modulo the fourth power of a cyclotomic polynomial from Watson's ${}_8\varphi _7$ transformation. Taking parameters substitution in our new q-congruence, we present a q-analogue of the (D.2) supercongruence of Van Hamme and several different q-analogues of He's supercongruence for $p\equiv 1(\mathrm{mod}3)$. Finally, we propose a Dwork-type supercongruence conjecture on He's supercongruence.

Introduction

In 1997, Van Hamme [27] developed several p-adic analogues of Ramanujan-type formulas for primes p. For example, the infinity series

$$\sum _{k=0}^{\infty }(4k+1)\frac{{(\frac{1}{2})}_k^4}{k{!}^4}=\infty ,\sum _{k=0}^{\infty }(6k+1)\frac{{(\frac{1}{3})}_k^6}{k{!}^6}=1.01226\cdots ,$$

have the following nice p-adic analogues

$$(\text{C.2})\sum _{k=0}^{(p-1)/2}(4k+1)\frac{{(\frac{1}{2})}_k^4}{k{!}^4}\equiv p(\mathrm{mod}{p}^3)\text{for }p>2,$$

$$(\text{D.2})\sum _{k=0}^{(p-1)/3}(6k+1)\frac{{(\frac{1}{3})}_k^6}{k{!}^6}\equiv -p{\mathrm{\Gamma }}_p{(1/3)}^9(\mathrm{mod}{p}^4)\text{for }p\equiv 1(\mathrm{mod}6).$$

Here ${(a)}_n=a(a+1)\cdots (a+n-1)$ is the Pochhammer symbol and ${\mathrm{\Gamma }}_p(x)$ denotes the p-adic Gamma function [23]. Van Hamme [27] himself proved (1.1) and the modulus $p^2$ case of (1.2). In 2011, Long [21] established the following generalization of (1.1):

$$\sum _{k=0}^{(p-1)/2}(4k+1)\frac{{(\frac{1}{2})}_k^4}{k{!}^4}\equiv p(\mathrm{mod}{p}^4);$$

and another interesting supercongruence:

$$\sum _{k=0}^{(p-1)/2}(4k+1)\frac{{(\frac{1}{2})}_k^6}{k{!}^6}\equiv p\sum _{k=0}^{(p-1)/2}\frac{{(\frac{1}{2})}_k^4}{k{!}^4}(\mathrm{mod}{p}^4).$$

Later, Long and Ramakrishna [22, Theorem 2] generalized Van Hamme's conjecture (D.2) to modulus $p^6$ case. They [22, Proposition 25] also gave the following supercongruences

$$\sum _{k=0}^{p-1}\frac{{(\frac{1}{3})}_k^3}{k{!}^3}\equiv \{\begin{array}{cc}{\mathrm{\Gamma }}_p{(1/3)}^6(\mathrm{mod}{p}^3),\hfill & \text{if }p\equiv 1(\mathrm{mod}6)\text{;}\hfill \\ -\frac{p^2}{3}{\mathrm{\Gamma }}_p{(1/3)}^6(\mathrm{mod}{p}^3),\hfill & \text{if }p\equiv 5(\mathrm{mod}6).\hfill \end{array}$$

In addition, He [15, Corollary 1.3] obtained the following Ramanujan-type supercongruences: for $p\ge 5$,

$$\sum _{k=0}^{(p-1)/3}(6k+1)\frac{{(\frac{1}{3})}_k^4}{k{!}^4}\equiv -p{\mathrm{\Gamma }}_p{(1/3)}^3(\mathrm{mod}{p}^4),\text{if }p\equiv 1(\mathrm{mod}3),$$

$$\sum _{k=0}^{(2p-1)/3}(6k+1)\frac{{(\frac{1}{3})}_k^4}{k{!}^4}\equiv 2p{\mathrm{\Gamma }}_p{(1/3)}^3(\mathrm{mod}{p}^4),\text{if }p\equiv 2(\mathrm{mod}3).$$

During the past few years, finding q-analogues of known congruences or supercongruence has become an important thing for more and more scholars. Especially, most of Van Hamme's supercongruences have been generalized to the q-world with various techniques [2], [3], [4], [5], [6], [7], [8], [10], [11], [12], [13], [30], [31]. Some other recent progress on q-congruences can be found in [9], [14], [16], [17], [18], [19], [20], [24], [25], [28], [29], [32]. For example, Guo and Schlosser [11, Theorem 2.2] proposed the following partial q-analogue of (1.4),

$$\sum _{k=0}^{(n-1)/2}[4k+1]\frac{{(q;q^2)}_k^6}{{(q^2;q^2)}_k^6}\equiv [n]q^{(1-n)/2}\sum _{k=0}^{(n-1)/2}\frac{{(q;q^2)}_k^4}{{(q^2;q^2)}_k^4}{q}^{2k},(\mathrm{mod}[n]{\mathrm{\Phi }}_n{(q)}^2),$$

where n is a positive odd integer, and Guo and Wang [12] reproved (1.3) by finding the following q-congruence

$$\sum _{k=0}^{(n-1)/2}[4k+1]\frac{{(q;q^2)}_k^4}{{(q^2;q^2)}_k^4}\equiv [n]q^{(1-n)/2}+{[n]}^3{q}^{(1-n)/2}\frac{(n^2-1){(1-q)}^2}{24}(\mathrm{mod}[n]{\mathrm{\Phi }}_n{(q)}^3).$$

By a result of Guo and Schlosser [10, Theorem 1.1], for $n\equiv 2(\mathrm{mod}3)$, we have

$$\sum _{k=0}^{(2n-1)/3}[6k+1]\frac{{(q;q^3)}_k^4}{{(q^3;q^3)}_k^4}{q}^k\equiv q^{(1-2n)/3}\frac{{(q^2;q^3)}_{(2n-1)/3}}{{(q^3;q^3)}_{(2n-1)/3}}[2n](\mathrm{mod}[n]{\mathrm{\Phi }}_n{(q)}^3),$$

which is a q-analogue of He's supercongruence (1.7).

Here and in what follows, for an indeterminate q, ${(a;q)}_n=(1-a)(1-aq)\cdots (1-a{q}^{n-1})$ is the q-shifted factorial and ${(a_1,a_2,\dots ,a_m;q)}_n={(a_1;q)}_n{(a_2;q)}_n\cdots {(a_m;q)}_n$ denotes a product of q-shifted factorials, $[n]={[n]}_q=(1-q^n)/(1-q)$ is the q-integer. Moreover, the n-th cyclotomic polynomial ${\mathrm{\Phi }}_n(q)$ is defined as the following product

$${\mathrm{\Phi }}_n(q)=\prod _{\begin{array}{c}\hfill 1⩽k⩽n\hfill \\ \hfill \gcd (n,k)=1\hfill \end{array}}(q-{\zeta }^k),$$

where ζ is an n-th primitive root of unity.

Inspired by the above work, we shall propose a new q-congruence modulo the fourth power of a cyclotomic polynomial, from which we can deduce a q-analogue of the (D.2) supercongruence of Van Hamme and a q-analogue of He's supercongruence (1.6).

The rest of the paper is organized as follows. We will give our main results in the next section. Then the proofs of our q-congruences will be given in Sections 3 and 4 by combining Guo and Zudilin's ‘creative microscoping’ method with the Chinese remainder theorem for coprime polynomials.