Proof of a Dwork-type supercongruence by induction

1.   Introduction

In the spirit of ^[25], the supercongruence

corresponds to a divergent Ramanujan-type series for $1/\pi$. In ^[8], Guillera and Zudilin made use of the Wilf-Zeilberger algorithmic technique to show that

Note that (1.2) is equivalent to (1.1) since $p\mid \binom{2k}{k}$ for $\frac{p+1}{2}\leq k\leq p-1$. The supercongruence (1.2) was independently observed numerically by Sun [21,Conjecture 5.1 (ii)]).

Given a prime $p$ and a positive integer $k$, for any positive integer $n$, we call a congruence as a Dwork-type congruence on a sequence $\{a_n\}_{n\geq 0}$ of integers if

where $\nu_p(n)$ is the largest integer such that $p^{\nu_p(n)}\mid n$. We refer the reader to ^[2,3,5,14] for Dwork-type congruences and ^[9,11,16,22] for $q$-analogues of Dwork-type congruences.

For any odd prime $p$ and integer $r\geq 1$, Guo ^[10] gave a $q$-analogue of (1.2) and made the following Dwork-type supercongruence conjecture:

and

where the Kronecker delta function is defined as $\delta_{m, n} = 1$ if $m = n$ and $\delta_{m, n} = 0$ otherwise. Guo and Zudilin ^[11] proved (1.3) and (1.4) modulo $p^{3r}$ by establishing their $q$-analogues.

Recently, Sun ^[20] conjectured that, for any prime $p > 3$ and positive integers $n, r$,

where $B_{n}$ are the Bernoulli numbers which can be defined by

and

where $E_{n}$ are the Euler numbers, i.e.,

and $(\frac{\cdot}{p})$ denotes the Legendre symbol. Clearly (1.3) and (1.4) follow from (1.5) and (1.6) by taking $n = 1$, respectively. In this paper, we shall prove the following result, which asserts that (1.3) is true modulo $p^{3r+1-\delta_{p, 3}}$.

Theorem 1.1 Let $p\geq 3$ be a prime and $r$ be a positive integer. Then

The remainder of the paper is organized as follows. In the next section, we are going to show some basic lemmas for Theorem 1.1. The proof of Theorem 1.1 will be given in Section 3.

2.   Some basic lemmas

From now on, for an assertion $A$ we adopt the notation:

We know that $[m = n]$ coincides with the Kronecker delta function $\delta_{m, n}$. For positive integers $n$ and $m$, the generalized harmonic numbers of order $m$ are defined by $H_{n, m} = \sum_{j = 1}^{n}\frac{1}{j^m}$. Clearly, $H_{n, 1} = H_n = \sum_{j = 1}^{n}\frac{1}{j}$. For convenience, we always abbreviate the Harmonic number $H_{k\in V, p\nmid k} = \sum_{k\in V, p\nmid k}\frac{1}{k}$ as $H_{k\in V}'$.

The following lemma is a result of Beukers [3,Lemma 2 (i)].

Lemma 2.1 Let $n, k, r$ be positive integers and $p$ be a prime. Then

Proof. Note that

This proves (2.1).

The following result is due to Gy [12,Lemma 2.7].

Lemma 2.2 Let $n\geq 1$ be an integer and $p$ be an odd prime such that $p > \frac{n+1}{2}$. We have the following congruence:

where $q_r(a) = \frac{a^{\phi(r)}-1}{r}$ denotes Euler's quotient of an integer $r$ with base $a$ and $\phi(r)$ is Euler's totient function.

The following lemma involving Bernoulli numbers is due to Lin [13,Corollary 4.3].

Lemma 2.3 Let $p$ be an odd prime and $j$ be a positive integer with $j\leq p-2$. Assume that $(2, d) = 1$. Then, we have

We also need three more lemmas.

Lemma 2.4 For primes $p$ and positive integers $a, b, r, s$, we have

The congruence (2.3) is called Jacobsthal's binomial congruence. Gessel ^[6], Granville ^[7] and Straub ^[18] gave extensions of (2.3) to nonnegative integers $a, b$ and negative integers $a, b$ independently.

Lemma 2.5 Let $p\geq3$ be an odd prime and $l$ be a nonnegative integer. For any positive integer $s$, we have

Proof. The case $p\geq 5$ in (2.4) and the case $p = 3$ in (2.5) are due to Beukers [3,Lemma 2.1] and Cai [4,Lemma 1], respectively. By (2.5),

we get

By (2.4) and (2.9), we immediately get

So (2.6) is valid. With the help of (2.5) and (2.9), we have

This proves (2.7).

Finally, Cai [4,Theorem 1] showed that, for any odd integer $n > 1$,

The congruence (2.8) is just the $n = p$ case of the above congruence.

Lemma 2.6 Let $p\geq 5$ be a prime. Then

Proof. Using the principle of mathematical induction, Alzer et al. ^[1] showed that

Note that

and

for any positive integer $k\leq p-1$ (see Sun [19,Lemma 2.1]). Using

and

the congruence (2.11) follows from

by (2.8) and (2.9).

3.   Proof of Theorem 1.1

We start with the following identity due to Mao and Zhang ^[15]:

Setting $n = p^{r-1+j}$ for $j\in\{0, 1\}$ in (3.1) yields

since for any odd prime $p$, we have $\binom{2p^{r}}{p^{r}}\equiv\binom{2p^{r-1}}{p^{r-1}}\pmod{p^{3r-\delta_{p, 3}}}$ by (2.3). In view of [23,Lemma 2.3], for any odd prime $p$ and positive integers $r, s$ with $r\geq s$, we get

It follows that

by Euler's theorem. For any integer $l\in\{0, 3^{r-1}-1\}$,

Clearly, $\binom{12}{6}\equiv\binom{4}{2}\pmod{27}$. As in ^[17], we know that the super Catalan numbers $S(m, n)$ are defined by $\frac{(2m)!(2n)!}{(m+n)!m!n!}$ and

is an integer. With the help of Lemmas 2.1, 2.4, 2.5 and the above congruences, we obtain

It follows that

by Lemma 2.5. Noting that

by Lemma 2.1, and

we have

In view of [24,Lemma 2.4] and (2.9), for any integer $l$ and nonnegative integer $s$, we get

and

By (2.9),

It follows that

and

For any nonnegative integer $s$, by (2.5), (2.12), (3.8) and (3.9), we have

For any positive integer $k\leq p-1$, we have $\binom{p-1}{k}(-1)^k\equiv{1-pH_k}\pmod{p^{2}}$. By (2.5) and (2.8),

In view of (2.5), (2.12) and Lemma 2.6, we have

Combining (3.2), (3.4), (3.6) with the above results, we get

Clearly, for any nonnegative integer $l$ and positive integer $s$, we have

by (2.6). Now we will handle the case $r = 1$ of (3.10), that is to say, $s = 0$. With the help of (2.5), (2.8) and (3.11), we obtain

The congruence (1.7) with $r = 1$ follows from (3.7), (3.10) and (3.12). Next suppose that $r\geq 2$. By (2.5), (2.7), (2.8), (3.3), (3.11) and Lemma 2.2 with $n = 2$, we have

For any positive integer $s\leq r$, by Lemma 2.1 and (3.7), we get

In view of (3.5),

By (2.6), for any positive integer $s\geq 2$, we have

Combining (2.5), (2.7), (3.11) with (3.14) gives

In the case $1\leq s\leq r-1$, with the help of (2.10) with $n = p^s$ and (3.3),

Note that

by (3.15) and (3.16) for any integer $1\leq s\leq r-1$. In light of Lemma 2.1, we obtain

By (3.13) and the above two congruences, for any integer $1\leq s\leq r-1$, we have

Combining (2.5), (3.7), (3.10), (3.17) with (1.7), by induction, it suffices to prove that

But this congruence immediately follows from (2.10) with $n = p^r$, (3.3), (3.11) and (3.14).

4.   Conclusions

The main result of this paper is a theorem. For any odd prime $p$ and positive integer $r$, the Dwork-type supercongruence (1.7) is proved, which is a extended result of Guo and Zudilin. Since Guo has made a conjecture that (1.7) is true modulo $p^{4r-\delta_{p, 3}}$, we hope that this work will open new inquiry opportunities in this fields for other researchers and knowledge seekers.

Acknowledgments

The work was partially supported by National Natural Science Foundation of China (Grant Nos. 11971222, 12071208 and 61502236) and Special Program for Student Innovation and Entrepreneurship Training (S20190025). We are grateful to the anonymous referees for useful comments on this paper. We also thank Professor V. J. W. Guo for helpful discussions.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.