Congruences for Ap\'ery numbers $\beta_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$

TL;DR

This paper investigates new congruences involving Apéry numbers, revealing deep number-theoretic properties and relations between different types of Apéry sequences, with implications for understanding their modular behavior.

Contribution

The paper establishes novel congruences for Apéry numbers and explores their relations, advancing the understanding of their arithmetic properties.

Findings

Proved a congruence involving Apéry numbers modulo 2n^2.

Derived a prime modulus congruence involving Bernoulli numbers.

Identified relations between two types of Apéry numbers.

Abstract

In this paper we establish some congruences involving the Ap\'ery numbers $\beta_{n}=\sum_{k=0}^{n}\binom{n}{k}^2\binom{n+k}{k}$ $(n=0,1,2,\ldots)$. For example, we show that $$\sum_{k=0}^{n-1}(11k^2+13k+4)\beta_k\equiv0\pmod{2n^2}$$ for any positive integer $n$, and $$\sum_{k=0}^{p-1}(11k^2+13k+4)\beta_k\equiv 4p^2+4p^7B_{p-5}\pmod{p^8}$$ for any prime $p>3$, where $B_{p-5}$ is the $(p-5)$th Bernoulli number. We also present certain relations between congruence properties of the two kinds of A\'pery numbers, $\beta_n$ and $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$.

Equations

$$\sum _{k=0}^{n-1}(11k^2+13k+4){\beta }_k\equiv 0(\mathrm{mod}2n^2)$$

$$\sum _{k=0}^{p-1}(11k^2+13k+4){\beta }_k\equiv 4p^2+4p^7{B}_{p-5}(\mathrm{mod}{p}^8)$$

$$A_n=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2{\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)}^2(n\in \mathbb{N}=\{0,1,2,\dots \}).$$

$${\beta }_n:=\sum _{k=0}^n{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)(n\in \mathbb{N}).$$

$$\sum _{k=0}^{n-1}(2k+1)A_k\equiv 0(\mathrm{mod}n)\text{for all}n\in {\mathbb{Z}}^{+}=\{1,2,3,\dots \}$$

$$\sum _{k=0}^{p-1}(2k+1)A_k\equiv p+\frac{7}{6}{p}^4{B}_{p-3}(\mathrm{mod}{p}^5)\text{for any prime}p>3,$$

$$B_0=1,\text{and}\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n+1}{k}\right)B_k=0\text{for all}n\in {\mathbb{Z}}^{+}.$$

$$\sum _{k=0}^{n-1}(2k+1)(-1{)}^k{A}_k\equiv 0(\mathrm{mod}n)\text{for all}n\in {\mathbb{Z}}^{+}$$

$$\sum _{k=0}^{p-1}(2k+1)(-1{)}^k{A}_k\equiv p\left(\frac{p}{3}\right)(\mathrm{mod}{p}^3)\text{for any prime}p>3,$$

$$\sum _{k=0}^{n-1}(6k^3+9k^2+5k+1)(-1{)}^k{A}_k\equiv 0(\mathrm{mod}{n}^3)\text{for all}n\in {\mathbb{Z}}^{+}.$$

$$\sum _{k=0}^{n-1}(11k^2+13k+4){\beta }_k\equiv 0(\mathrm{mod}2n^2),$$

$$\sum _{k=0}^{n-1}(11k^2+9k+2)(-1{)}^k{\beta }_k\equiv 0(\mathrm{mod}2n^2),$$

$$\sum _{k=0}^{n-1}(11k^3+7k^2-1)(-1{)}^k{\beta }_k\equiv 0(\mathrm{mod}{n}^2).$$

$$\sum _{k=0}^{p-1}(11k^2+13k+4){\beta }_k$$

$$\sum _{k=0}^{p-1}(11k^2+9k+2)(-1{)}^k{\beta }_k$$

$$\sum _{k=0}^{p-1}(11k^3+7k^2-1)(-1{)}^k{\beta }_k$$

$$\equiv$$

$$A_n\equiv 8j+4i+1(\mathrm{mod}16)$$

$${\beta }_n\equiv 2(i+j)+1(\mathrm{mod}4).$$

$$A_n=1+4\sum _{0<k⩽n}{\left(\genfrac{}{}{0ex}{}{n+k}{2k}\right)}^2{\left(\genfrac{}{}{0ex}{}{2k-1}{k-1}\right)}^2\equiv 1(\mathrm{mod}4).$$

$$A_n\equiv 16k+8j+4i+1(\mathrm{mod}32)\text{and}{\beta }_n\equiv 4m+2l+1(\mathrm{mod}8)$$

$$L_2(n)\equiv 4(k+m)+2j+i(\mathrm{mod}8),$$

$$\sum _{k=0}^{n-1}(11k^2+13k+4){\beta }_k=-\sum _{k=0}^{n-1}{\left(\genfrac{}{}{0ex}{}{n}{k+1}\right)}^2\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)a_1(n,k),$$

$$\sum _{k=0}^{n-1}(11k^2+9k+2)(-1{)}^{n-1-k}{\beta }_k=\sum _{k=0}^{n-1}{\left(\genfrac{}{}{0ex}{}{n}{k+1}\right)}^2(\genfrac{}{}{0ex}{}{n+k}{k})a_2(n,k),$$

$$\sum _{k=0}^{n-1}(11k^3+7k^2-1)(-1{)}^{n-1-k}{\beta }_k=\sum _{k=0}^{n-1}{\left(\genfrac{}{}{0ex}{}{n}{k+1}\right)}^2(\genfrac{}{}{0ex}{}{n+k}{k})a_3(n,k),$$

$$a_1(n,k)=$$

$$a_2(n,k)=$$

$$a_3(n,k)=$$

$$\sum _{k=0}^{n-1}(11k^2+13k+4){\beta }_k=\sum _{k=0}^{n-1}\sum _{l=0}^{n-1}F(k,l)$$

$$F(k,l)=(11k^2+13k+4){\left(\genfrac{}{}{0ex}{}{k}{l}\right)}^2\left(\genfrac{}{}{0ex}{}{k+l}{l}\right).$$

$$F_1(k,l)=-\frac{(k-l{)}^2{a}_1(k,l)}{(l+1{)}^2(11k^2+13k+4)}F(k,l)$$

$$F_2(k,l)=\frac{l(3kl-l^2+2k-2l)}{11k^2+13k+4}F(k,l)$$

$$F(k,l)=(F_1(k+1,l)-F_1(k,l))+(F_2(k,l+1)-F_2(k,l))$$

$$(11k^2+9k+2)(-1{)}^k{\left(\genfrac{}{}{0ex}{}{k}{l}\right)}^2(\genfrac{}{}{0ex}{}{k+l}{l})$$

Full text

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for any positive integer $n$, and

[TABLE]

for any prime $p>3$, where $B_{p-5}$ is the $(p-5)$th Bernoulli number. We also present certain relations between congruence properties of the two kinds of Aṕery numbers, $\beta_{n}$ and $A_{n}=\sum_{k=0}^{n}\binom{n}{k}^{2}\binom{n+k}{k}^{2}$.

2010 Mathematics Subject Classification. Primary 11B65; Secondary 11A07, 11B68, 11B75.

Keywords. Apéry numbers, Bernoulli numbers, congruences.

The research is supported by the NSFC-RFBR Cooperation and Exchange Program (NSFC Grant No. 11811530072, RFBR Grant No. 18-51-53020-GFENxa). The first and the third authors are also supported by the Natural Science Foundation of China (Grant No. 11971222).

1. Introduction

In 1979 R. Apéry (see [1] and [19]) established the irrationality of $\zeta(3)=\sum_{n=1}^{\infty}1/n^{3}$ by using the Apéry numbers

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His method also allowed him to prove the irrationality of $\zeta(2)=\pi^{2}/6$ via another kind of Apéry numbers

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In 2012, the third author [15] proved that

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and

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where $B_{0},B_{1},B_{2},\ldots$ are the Bernoulli numbers defined by

[TABLE]

(For the basic properties of Bernoulli numbers, see [8, pp. 228-248].) The third author also conjectured that

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and

[TABLE]

which was confirmed by V.J.W. Guo and J. Zeng [5]. In 2016 the third author [17] showed that

[TABLE]

In view of this, it is natural to ask whether the Apéry numbers \beta_{n}\ also have such kind of congruence properties. This is the main motivation of this paper.

Another motivation is due to a theorem of E. Rowland and R. Yassawi [12] about $A_{n}\ (n\in\mathbb{N})$ modulo $16$. We will extend their result to $A_{n}$ modulo $32$; quite unexpectedly, this requires consideration of $\beta_{n}$ modulo $8$.

Now we state our main results which were first found via Mathematica.

Theorem 1.1**.**

For any positive integer $n$, we have

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Theorem 1.2**.**

Let $p>3$ be a prime. Then

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and

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where $H_{p-1}$ denotes the harmonic number $\sum_{k=1}^{p-1}\frac{1}{k}$.

Theorem 1.3**.**

Let $n$ be a nonnegative integer. If

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with $\{i,j\}\subseteq\{0,1\}$, then

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Remark 1.4*.*

For any $n\in\mathbb{N}$, we clearly have

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Theorem 1.5**.**

Let $n$ be a nonnegative integer. If

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with $\{i,j,k,l,m\}\subseteq\{0,1\}$, then

[TABLE]

where $L_{2}(n)$ (the binary run-length of $n$) is defined as the number of blocks of consecutive [math]’s and $1$’s in the binary expansion of $n$.

We will prove Theorem 1.1 in the next section. We are going to provide some lemmas in Section 3 and show Theorem 1.2 in Section 4. Theorems 1.3 and 1.5 will be proved in Section 5.

2. Proof of Theorem 1.1

Lemma 2.1**.**

For any $n\in\mathbb{Z}^{+}$, we have

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where

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Proof. Note that

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is a double sum with

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Using the method in Mu and Sun’s paper [9], we find

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and

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so that

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which can be verified directly. This allows us to reduce the double sum $\sum_{k=0}^{n-1}\sum_{l=0}^{n-1}F(k,l)$ to the right-hand side of (2.1).

Identities (2.2) and (2.3) can be deduced similarly; in fact,

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and

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where

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This ends our proof. ∎

Lemma 2.2**.**

For any positive even integer $n$, we have

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Proof. Clearly,

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For $k\in\{0,\ldots,n-1\}$, if $2\mid k$ then $n+k$ is even and hence

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if $2\nmid k$ then

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Therefore,

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where we have applied (in the last step) the Chu-Vandermonde identity (cf. [4, (3.1)]). This proves (2.5). ∎

Proof of Theorem 1.1. For each $k=0,\ldots,n-1$, clearly

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and

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where $C_{k}$ is the Catalan number $\binom{2k}{k}/(k+1)=\binom{2k}{k}-\binom{2k}{k+1}$.

(i) As

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we have

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Clearly $11k^{2}+13k+4\equiv 0\ ({\rm{mod}}\ 2)$ for all $k=0,\ldots,n-1$. So, by Lemma 2.1 and the above, (1.1) holds when $2\nmid n$.

Now suppose that $n$ is even. By Lemma 2.1 and the above, we have

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With the help of the Chu–Vandermonde identity,

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and

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In view of (2.5), we also have

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since $kC_{k}=\binom{2k}{k+1}$. Therefore (1.1) holds.

(ii) As $a_{2}(n,k)=3n^{2}-(k+1)n$, we have

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Clearly $11k^{2}+9k+2\equiv 0\ ({\rm{mod}}\ 2)$ for all $k=0,\ldots,n-1$. So, by Lemma 2.1 and the above, (1.2) holds when $2\nmid n$.

Now suppose that $n$ is even. By Lemma 2.1 and the above, we have

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By the Chu–Vandermonde identity,

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Combining this with (2.6) we obtain (1.2).

(iii) Since

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we have

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and hence (1.3) follows from (2.3).

In view of the above, we have completed the proof of Theorem 1.1. ∎

3. Some lemmas

For each $s\in\mathbb{Z}^{+}$, we define

[TABLE]

and call such numbers harmonic numbers of order $s$. Those $H_{n}:=H_{n,1}\ (n\in\mathbb{N})$ are usually called harmonic numbers.

Lemma 3.1**.**

For any $n\in\mathbb{Z}^{+}$, we have

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Remark 3.2*.*

The identities (3.1)-(3.3) are known. See [4, (1.45)] for (3.1), [6] for (3.2), and the proof of [16, Lemma 3.1] for simple proofs of (3.2) and (3.3).

Let $p>3$ be a prime. In 1900 J.W.L. Glaisher [2, 3] refined Wolstenholme’s work [20] on congruences by showing that

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and

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Lemma 3.3**.**

(X. Zhou and T. Cai [22, p. 1332])* Let $r,s\in\mathbb{Z}^{+}$. For any prime $p>rs+2$, we have*

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Lemma 3.4**.**

(i) (J. Zhao [21, Theorems 3.1 and 3.2])* Let $s,t\in\mathbb{Z}^{+}$ and let $p\geqslant s+t$ be a prime. Then*

[TABLE]

If $s\equiv t\ ({\rm{mod}}\ 2)$ and $p>s+t+1$, then

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(ii) (J. Zhao [21, Theorem 3.5])* Let $r,s,t\in\mathbb{Z}^{+}$ with $r+s+t$ odd. For any prime $p>r+s+t$, we have*

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(iii) (J. Zhao [21, Corollary 3.6])* For any prime $p>5$, we have*

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Lemma 3.5**.**

For any prime $p>5$, we have

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and

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Proof. (i) Clearly,

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and hence

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Note that $H_{p-1,3}\equiv 0\ ({\rm{mod}}\ p^{2})$ by (3.5) or Lemma 3.3, and

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by R. Tauraso [18, Theorem 1.3]. Thus

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By Lemma 3.2,

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Combining these with (3.8), we immediately obtain (3.6).

(ii) As $H_{p-1}\equiv 0\ ({\rm{mod}}\ p)$ and

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we have

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with the help of Lemma 3.4 and (3.4). This proves (3.7).

The proof of Lemma 3.5 is now complete. ∎

Lemma 3.6**.**

Let $p>5$ be a prime. Then

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Proof. As

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for all $k=1,\ldots,p-1$, we have

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By (3.10),

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Note also that

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with the help of parts (i) and (iii) of Lemma 3.4. Combining the above, we get the desired congruence (3.11). ∎

Lemma 3.7**.**

For any prime $p>5$, we have

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Proof. It is known (cf. [18, Theorem 2.4]) that

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By Lemma 3.3,

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So, the desired congruence follows. ∎

Lemma 3.8**.**

For any prime $p>7$, we have

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Proof. Clearly,

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since $H_{p-1,6}\equiv 0\ ({\rm{mod}}\ p)$ and $H_{p-1,5}\equiv 0\ ({\rm{mod}}\ p^{2})$ by (3.5) or Lemma 3.3. Note that

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by [14, Theorem 5.1 and Remark 5.1]. Therefore

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and hence (3.13) follows. ∎

Remark 3.9*.*

Let $p>3$ be a prime. If $p>7$ then (3.13) has the following equivalent form

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in view of (3.12). It is easy to see that (3.14) also holds for $p=5,7$.

4. Proof of Theorem 1.2

To prove Theorem 1.2, we need an auxiliary theorem.

Theorem 4.1**.**

Let $p>5$ be a prime. Then

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and

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Proof.

For each $k=1,\ldots,p-1$, we clearly have

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and

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Let $r\in\{0,1,2\}$ and

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In view of the above, we have

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Case 1. $r=0$.

In this case,

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by (3.1), and hence

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with the help of Lemma 3.6. This proves (4.1).

Case 2. $r=1$.

In this case,

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and also

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by Lemma 3.4. Thus

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In view of (3.3) and (4.4), we have

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Note that

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by (3.14) and (3.12). In view of Lemma 3.4,

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and

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Therefore

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Combining this with (4.7), we see that

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This proves (4.2).

Case 3. $r=2$.

In this case,

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and hence

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with the help of (3.3), (3.14) and Lemma 3.5. This yields the desired (4.3).

In view of the above, we have completed the proof of Theorem 4.1. ∎

Proof of Theorem 1.2. It is easy to see that (1.4)-(1.6) hold for $p=5$. Below we assume $p>5$.

For $i=1,2,3$ let

[TABLE]

with $a_{i}(n,k)$ given by Lemma 2.1. In view of Lemma 2.1, it suffices to show the following three congruences:

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Clearly,

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Note that

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In view of Lemma 3.7, we have

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For $r=0,1,2$ let $\sigma_{r}$ be defined as in (4.6). Noting

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and applying Theorem 4.1, we get

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Combining this with (4.11) and (4.12), we get (4.8).

Similarly,

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This proves (4.9).

As

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we have

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Combining this with (4.11) and (4.12), we get (4.10).

The proof of Theorem 1.2 is now complete. ∎

5. Proofs of theorems 1.3 and 1.5

Clearly $L_{2}(0)=0$, and for each $n\in\mathbb{Z}^{+}$ we have

[TABLE]

E. Rowland and R. Yassawi discovered that the two least significant binary digits of  $L_{2}(n)$ are completely determined by the fourth and the third least significant digits of the Apéry number $A_{n}$. (As $A_{n}\equiv 1\ ({\rm{mod}}\ 4)$, the two least significant digits of $A_{n}$ are always [math] and $1$, and hence these digits contain no information about $n$.)

Theorem 5.1**.**

[12, Theorem 3.28]** Let $n\in\mathbb{N}$. If

[TABLE]

with $\{i,j\}\subseteq\{0,1\}$, then

[TABLE]

Theorem 1.3 shows that the fourth and the third least significant digits of $A_{n}$ also determine the second least significant digit of $\beta_{n}$ (all numbers $\beta_{n}$ are odd, so the very least significant digit is always $1$).

Theorem 1.5 shows that the knowledge of the five least significant digits of $A_{n}$ by itself is not sufficient for determining the three least significant digits of $L_{2}(n)$. This is because the third least significant digit of $\beta_{n}$ can be [math] or $1$. But the additional knowledge of this digit gives the missing bit of information for determining the three least significant digits of $L_{2}(n)$ according to Theorem 1.5.

Theorems 1.3 and 1.5 can be proved by the technique used in [12] with the aid of software [10] implemented by E. Rowland.

Namely, Apéry numbers of both kinds are known to be the diagonal sequences of certain rational functions. In particular, A. Straub [13] proved that $A_{n}$ is the coefficient of $x_{1}^{n}x_{2}^{n}x_{3}^{n}x_{4}^{n}$ in the formal Taylor expansion of the function

[TABLE]

Similar coefficient in the expansion of

[TABLE]

is, according to [12], equal to $\beta_{n}$.

According to [12, Theorem 2.1], the above representations of Apéry numbers as the diagonal coefficients imply that for every prime $p$ and its power $q$ the sequences $A_{n}\!\!\mod{q}$ and $\beta_{n}\!\!\mod{q}$ are $p$-automatic. Moreover, [12] contains two algorithms for constructing corresponding automata; Mathematica implementations of these algorithms are given in [10].

To prove Theorem 1.3, we need to examine the sequence $A_{n}\!\!\mod{16}$. Corresponding automaton $\mathcal{A}_{16}$ was calculated in [11] and exhibited in [12], we reproduce it here in Figure 1 (with the initial state marked in light color). This automaton calculates $A_{n}\!\!\mod{16}$ in the following way. Let

[TABLE]

where $\{n_{0},\dots,n_{m}\}\subseteq\{0,1\}$; start from the initial state and follow the oriented path formed by edges marked $n_{0},\dots,n_{m}$; the final vertex is labeled by $A_{n}\!\!\mod{16}$.

Function AutomaticSequenceReduce from [10], being applied to (5.4), returns the minimal automaton $\mathcal{B}_{4}$ calculating $\beta_{n}\!\!\mod{4}$; this automaton is exhibited in Figure 2.

Let

[TABLE]

Theorem 1.3 asserts that

[TABLE]

We can construct an automaton calculating $f(A_{n}\!\!\mod{16})$ simply by applying function $f$ to the labels of states in automaton $\mathcal{A}_{16}$. The resulting automaton is not minimal, but we can minimize it using, for example, function AutomatonMinimize from [10]. The minimal automaton calculating $f(A_{n}\!\!\mod{16})$ turns out to be equal to $\mathcal{B}_{4}$, which proves (5.6) and hence Theorem 1.3 is proved as well.

The minimal automaton $\mathcal{A}_{32}$ calculating $A_{n}\!\!\mod{32}$ was constructed in [11], it has 33 states; the minimal automaton $\mathcal{B}_{8}$ calculating $\beta_{n}\!\!\mod{8}$ has 17 states. Taking into account Theorem 5.1, for proving Theorem 1.5 it is sufficient to work just with the fifth least significant digits of $A_{n}$ and the third least significant digits of $\beta_{n}$ and $L_{2}(n)$, which allows us to deal with automata having fewer number of states.

Figures 3 and 4 exhibit the minimal automata $\mathcal{A}_{32,5}$ and $\mathcal{B}_{8,3}$ calculating respectively the fifth least significant digit of $A_{n}$ and the third least significant digit of $\beta_{n}$. These automata are easily constructed from $\mathcal{A}_{32}$ and $\mathcal{B}_{8}$ by properly relabelling the states and minimizing resulting automata.

With function AutomatonProduct from [10] we can easily construct automaton $\mathcal{C}$ which is the product of $\mathcal{A}_{32,5}$ and $\mathcal{B}_{8,3}$. Each state of $\mathcal{C}$ is labeled by two bits (corresponding to the fifth digit of $A_{n}$ and the third digit of $\beta_{n}$). Replacing each such pair of bits by their sum modulo $2$ and performing minimization, we get the automaton exhibited on Figure 5. It is not difficult to see that this automaton calculates the third digit of $L_{2}(n)$, which proves Theorem 1.5.

Acknowledgment. The authors would like to thank the third author’s PhD student Chen Wang for helpful comments on Lemmas 2.1 and 3.8.