Some Families of Super Congruences Involving Alternating Multiple Harmonic Sums

TL;DR

This paper investigates super congruences involving alternating multiple harmonic sums, extending previous results to broader families and providing new insights into their properties modulo prime powers.

Contribution

It introduces new families of super congruences involving alternating sums, generalizing earlier specific cases for n=4 and 5.

Findings

Extended super congruences for alternating sums modulo prime powers.

General formulas that encompass previous special cases.

Deeper understanding of the structure of alternating multiple harmonic sums.

Abstract

Let $p$ be a prime. In this short note we study some families of super congruences involving the following alternating sums \begin{equation*} \sum_{\substack{j_1+j_2+\cdots+j_n=2 p^r p\nmid j_1 j_2 \cdots j_n}} \frac{(-1)^{j_1+\cdots+j_b}}{j_1\cdots j_n} \pmod{p^r}, \end{equation*} which extend similar statements proved by Shen and Cai who treated the cases when $n=4,5$.

Equations

$$\sum _{\begin{array}{c}{j}_1+j_2+\cdots +j_n=2p^r\\ p\nmid j_1{j}_2\cdots j_n\end{array}}\frac{(-1{)}^{j_1+\cdots +j_b}}{j_1\cdots j_n}(\mathrm{mod}{p}^r),$$

$${\mathcal{H}}_n(\mathbf{s}):=\sum _{n>k_1>\cdot >k_d>0}\frac{1}{k_1^{s_1}\dots k_d^{s_d}}.$$

$$\sum _{\begin{array}{c}i+j+k=p\\ i,j,k\ge 1\end{array}}\frac{1}{ijk}\equiv -2B_{p-3}(\mathrm{mod}p),$$

$$\frac{t}{e^t-1}=\sum _{k=0}^{\infty }{B}_k\frac{t^k}{k!}.$$

$$Z_n(N,p):=$$

$$R_n^{(m)}(p^r):=$$

$$S_n^{(m)}(p^r):=$$

$${\sigma }_n^{(b)}(N,p):=\sum _{\begin{array}{c}{l}_1+l_2+\cdots +l_n=N\\ l_1,\dots ,l_n\in {\mathcal{P}}_p\end{array}}\frac{(-1{)}^{l_1+\cdots +l_b}}{l_1{l}_2\dots l_n}\text{for }p|N.$$

$$Z_n(N,p)=\frac{n!}{N}\sum _{\begin{array}{c}1\le u_1<\cdots <u_{n-1}<N\\ u_1,u_2-u_1,\dots ,u_{n-1}-u_{n-2},u_{n-1}\in {\mathcal{P}}_p\end{array}}\frac{1}{u_1\dots u_{n-1}}.$$

$$Z_n(N,p)=$$

$$=$$

$$\frac{1}{l_1\dots l_{n-1}}=\frac{l_1+\cdots +l_{n-1}}{l_1\dots l_{n-1}(l_1+\cdots +l_{n-1})}$$

$$Z_n(N,p)=\frac{n(n-1)}{N}\sum _{\begin{array}{c}{l}_1+\cdots +l_{n-2}<u_{n-1}<N\\ l_1,\dots ,l_{n-2},u_{n-1}-l_1-\cdots -l_{n-2},N-u_{n-1}\in {\mathcal{P}}_p\end{array}}\frac{1}{l_1{l}_2\dots l_{n-2}{u}_{n-1}},$$

$$S_n^{(m)}(p^{r+1})\equiv (-1{)}^{m-1}(\genfrac{}{}{0ex}{}{n-2}{m-1})S_n^{(1)}(p^2)p^{r-1}(\mathrm{mod}{p}^{r+1}).$$

$${\gamma }_n(a):=(-1{)}^{a+1}\frac{(a-1)!(n-1-a)!}{(n-1)!}.$$

$$S_n^{(m)}(p^{r+1})$$

$$\equiv (-1{)}^{m-1}(\genfrac{}{}{0ex}{}{n-2}{m-1})S_n^{(1)}(p^{r+1})$$

$$S_n^{(1)}(p^{r+1})\equiv p{S}_n^{(1)}(p^r)(\mathrm{mod}{p}^{r+1})$$

$$R_n^{(m)}(p)\equiv$$

$$R_n^{(m)}(p^r)\equiv$$

$$l_i=x_i{p}^r+y_i,x_i\ge 0,1\le y_i<p^r,y_i\in {\mathcal{P}}_p,1\le i\le n.$$

$$\Big{(}\sum_{i=1}^{n}x_{i}\Big{)}p^{r}+\sum_{i=1}^{n}{y_{i}}=mp^{r},$$

$$\left\{\begin{array}[]{ll}x_{1}+\cdots+x_{n}=m-a,\\ y_{1}+\cdots+y_{n}=ap^{r}.\\ \end{array}\right.$$

$$R_n^{(m)}(p^r)$$

$$=\sum _{a=1}^{n-1}\sum _{\begin{array}{c}{x}_1+\cdots +x_n=m-a\\ y_1+\cdots +y_n=a{p}^r\\ y_i\in {\mathcal{P}}_p,y_i<p^r\end{array}}\frac{1}{(x_1{p}^r+y_1)\cdots (x_n{p}^r+y_n)}$$

$$\equiv \sum _{a=1}^{n-1}\left(\genfrac{}{}{0ex}{}{m+n-a-1}{n-1}\right)S_n^{(a)}(p^r)$$

$$\equiv \sum _{a=1}^{n-1}\left(\genfrac{}{}{0ex}{}{m+n-a-1}{n-1}\right)(-1{)}^{a-1}(\genfrac{}{}{0ex}{}{n-2}{a-1})S_n^{(1)}(p^2)p^{r-2}$$

$$\sum _{a=1}^{n-1}(-1{)}^{a-1}(\genfrac{}{}{0ex}{}{n-2}{a-1}\left)\right(\genfrac{}{}{0ex}{}{m+n-a-1}{n-1})$$

$$=$$

$${\mathcal{H}}_n(s_1,\dots ,s_d):=\sum _{n>k_1>\cdot >k_d>0}\frac{\mathrm{sgn}(s_1{)}^{k_1}\cdots \mathrm{sgn}(s_d{)}^{k_d}}{k_1^{|s_1|}\dots k_d^{|s_d|}}.$$

$${\sigma }_n^{(b)}(N,p)=\sum _{\begin{array}{c}{l}_1+l_2+\cdots +l_n=N\\ l_1,\dots ,l_n\in {\mathcal{P}}_p\end{array}}\frac{(-1{)}^{l_1+\cdots +l_b}}{l_1{l}_2\dots l_n}\text{for }p|N.$$

$$F_a^{(b)}(\mathbf{s},N,p):=$$

$$N{\sigma }_n^{(b)}(N,p)=$$

$$F_a^{(b)}(\mathbf{s},N,p)=$$

$$X_m:=(1_m),Z_n:=(\{\bar{1}\},1_{n-1}).$$

Full text

[TABLE]

which extend similar statements proved by Shen and Cai who treated the cases when $n=4,5$. Our method works for arbitrary $n$.

1. Introduction

Over the past quarter of a century, multiple zeta values (MZVs) and their various generalizations have been intensively studied by many mathematicians and physicists due to their important applications in quite a few different areas of mathematics and theoretical physics. These values are infinite series whose finite sums are commonly called the multiple harmonic sums, defined as follows. Let ${\mathbb{N}}$ and ${\mathbb{N}}_{0}$ be the set of positive integers and nonnegative integers, respectively. For any $n,d\in{\mathbb{N}}$ and ${\bf s}=(s_{1},\dots,s_{d})\in{\mathbb{N}}^{d}$, we define the multiple harmonic sums (MHSs) by

[TABLE]

For example, ${\mathcal{H}}_{n+1}(1)$ is often called the $n$th harmonic number.

Very recently, a finite version of MZVs has emerged which has been conjectured to be closely related to MZVs, see [15, Ch. 8]. These values are essentially the MHSs truncated at different primes and then taken residues modulo the corresponding primes. Such congruences were first studied independently by the last author in [12, 13] and Hoffman in [3]. In general, it is well-known that Bernoulli numbers play a very important role in these congruences, see [8] for some classical results. As an application, in [11] the last author proved, by using some special properties of the double harmonic sums, that for every odd prime $p$

[TABLE]

where $B_{k}$ are Bernoulli numbers defined by the generating series

[TABLE]

Later, Ji gave an alternative simpler proof of (1) in [4] using some combinatorial techniques. Congruence (1) has since been generalized by either increasing the number of indices, changing the bound from $p$ to multiples of $p$ or $p$-powers, and/or considering the corresponding supercongruences (see [1, 5, 9, 10, 14, 16]), or even allowing the alternating version of MHSs (see [6, 7]).

Our main results of this short note concern the following type of sums. Let ${\mathcal{P}}_{p}$ be the set of positive integers not divisible by $p$. For $m,n,r,N\in{\mathbb{N}}$, we define

[TABLE]

The primary goal of our study is to find nice and simple supercongruences involving alternating sums defined as follows:

[TABLE]

We will reduce these congruences to those of $Z_{n}(N,p)$ whose special cases $R_{n}^{(m)}(p^{r})$ are closely related to $S_{n}^{(m)}(p^{r})$ by Proposition 2.3. These results are motivated by the recent work of Shen and Cai [6] who studied the above sums for $n=3,4$. In Theorem 3.4 we generalize this to arbitrary $n$ by using $R^{(m)}_{n}(p^{r})$ with $m=1,2$.

Acknowledgement. We would like to thank the anonymous referee for the careful reading of the manuscript and helpful comments and suggestions.

2. Some useful lemmas

We start with a formula expressing the sums $Z_{n}(N,p)$ in terms of a modified version of multiple harmonic sums.

Lemma 2.1**.**

Let $n,N\in{\mathbb{N}}$ and $p$ be a prime. If $p|N$ then we have

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Proof.

First, noting that $l_{1}+l_{2}+\dots+l_{n}=N$, we have

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Then one writes

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to get

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and continues in this way by using the substitutions $u_{j}=l_{1}+\dots+l_{j}$ for $1\leq j<n$ to prove equation (2). This completes the proof of the lemma. ∎

Lemma 2.2**.**

Suppose $m,n,r\in{\mathbb{N}}$ and $p$ is a prime with $p>n+1$. Then we have

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Proof.

For all $n,a\in{\mathbb{N}}$, set

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By [5, Lemma 2.3], we have

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So the lemma follows from [5, (1.3)] which says

[TABLE]

for all $r\geq 2$. ∎

Proposition 2.3**.**

Let $m,n,r\in{\mathbb{N}}$ with $r\geq 2$. Then we have

[TABLE]

Proof.

Let $p$ be a prime number such that $p>n+1$. For any $n$-tuples $(l_{1},\cdots,l_{n})$ of integers in ${\mathcal{P}}_{p}$ satisfying $l_{1}+\cdots+l_{n}=mp^{r}$, we rewrite

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Since

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there exists $1\leq a<n$ such that

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For $1\leq a<n$, the equation $x_{1}+\cdots+x_{n}=m-a$ has $\binom{m+n-a-1}{n-1}$ nonnegative integer solutions. Hence, for all $r\geq 1$,

[TABLE]

by Lemma 2.2. Note that the penultimate step holds for $r=1$ which implies (3). However, the last step is valid only when $r\geq 2$. So (4) follows immediately from

[TABLE]

by the famous Chu–Vandermonde identity. ∎

3. Alternating sums

We now define the alternating version of the multiple harmonic sums. For convenience, we denote by $\bar{s}$ a signed integer for every $s\in{\mathbb{N}}$ and set $|\bar{s}|=s$ and $\operatorname{{sgn}}(\bar{s})=-1$. Let $s_{j}$ be either a positive integer or a signed integer for all $j=1,\dots,d$. For any $n\in{\mathbb{N}}$, the alternating MHS is defined by

[TABLE]

For example, $\lim_{n\to\infty}{\mathcal{H}}_{n}(\bar{1})$ is just the well-known alternating harmonic series.

As variations of alternating MHSs, we have defined that

[TABLE]

In this section, for each fixed $n\geq 4$, we will study some suitable linear combinations of ${\sigma}_{n}^{(b)}(N,p)$ for $b=1,\dots,n-1$. To this end, for any $a\geq b\geq 0$, $d\geq 0$ and ${\bf s}=(s_{1},\dots,s_{d})\in\{1,\bar{1}\}^{d}$, we define

[TABLE]

Then it is easy to see that if $N$ is even then

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Here, we have abused the notation by writing $({\bf s},1)=(s_{1},\dots,s_{d},1)$ and $({\bf s},\bar{1})=(s_{1},\dots,s_{d},\bar{1})$. For $m\in{\mathbb{N}}_{0}$ and $n\in{\mathbb{N}}$, put

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For ${\bf s}=(X_{w_{1}},Z_{w_{2}},\dots,Z_{w_{l}})$, we set $W_{\bf s}:=(w_{1},w_{2},\dots,w_{l})$, $\operatorname{{len}}(W_{\bf s}):=l$ and

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Otherwise, for ${\bf s}\neq\emptyset$, we define

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Finally, for all fixed $a\geq b\geq 0$ and $A,B\geq 0$, we put

[TABLE]

where $(x)_{\alpha}=x(x-1)\cdots(x-{\alpha}+1)$ is the Pochhammer symbol for the falling factorial.

Lemma 3.1**.**

Let $a,b\in{\mathbb{N}}_{0}$. Then for any fixed nonnegative integer $d\leq a$,

[TABLE]

Proof.

We will prove this by induction on $d$. If $d=0$ then there is only one term in the sum corresponding to ${\bf s}=\emptyset$. Then the lemma holds by (7). Now let $d\geq 1$ and suppose the lemma is true when $d$ is replaced by $d-1$. Observe that any composition in $\{1,\bar{1}\}^{d}$ is produced by either $({\bf s},1)$ or $({\bf s},\bar{1})$ for a unique ${\bf s}\in\{1,\bar{1}\}^{d-1}$. Further, it is easy to see that

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If $d<a$ and $2\nmid\operatorname{{len}}(W_{\bf s})$, then by (6)

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If $d<a$ and $2|\operatorname{{len}}(W_{\bf s})$, then by (6) again

[TABLE]

This finishes the induction proof of the lemma. ∎

Corollary 3.2**.**

Let $a,b\in{\mathbb{N}}$ with $a\geq b$. For all ${\bf s}\in\{1,\bar{1}\}^{a}$, we have

[TABLE]

Proof.

It is easy to see that $A_{\bf s}+B_{\bf s}=|W_{\bf s}|=|{\bf s}|=a$. If $C_{a,b}(A_{\bf s},B_{\bf s})\neq 0$, then by its definition

[TABLE]

which imply that $A_{\bf s}=a-b,B_{\bf s}=b$ and $C_{a,b}(A_{\bf s},B_{\bf s})=(a-b)!b!$. ∎

Corollary 3.3**.**

For all fixed $a\in{\mathbb{N}}$, we have

[TABLE]

Proof.

By Corollary 3.2, $C(A_{\bf s},B_{\bf s})\neq 0$ for one and only one $b$ for every ${\bf s}\in\{1,\bar{1}\}^{a}$. Thus,

[TABLE]

As the term is nonzero only when all indices are even, we get

[TABLE]

We can now finish the proof of the corollary by applying Lemma 2.1. ∎

Theorem 3.4**.**

Let $n,N$ be two positive integers and $p$ a prime. If $p|N$ then we have

[TABLE]

where ${\alpha}_{n,b}=1$ except for ${\alpha}_{n,n/2}=1/2$ when $n$ is even. In particular, for every $r\in{\mathbb{N}}$ and prime $p$ we have

[TABLE]

Proof.

For even $N$, we have ${\sigma}_{n}^{(b)}(N,p)={\sigma}_{n}^{(n-b)}(N,p)$ and therefore we get

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by (5). Using substitution $b\to n-b$ in the second sum, we get

[TABLE]

by Corollary 3.3 with $a=n-1$. Therefore,

[TABLE]

since ${\sigma}_{n}^{(0)}(2N,p)=Z_{n}(2N,p)$. The final congruence of the theorem follows easily from Proposition 2.3. This completes the proof of the theorem. ∎

Corollary 3.5**.**

Let $n\in{\mathbb{N}}$ and $p$ be a prime such that $p>n+1$. Then we have

[TABLE]

Proof.

This follows easily from Theorem 3.4, [16, Main Theorem], [5, Lemma 3.5 and Corollary 3.6] (for $n$ odd) and [10, Theorem 1 and Corollary 1] (for $n$ even). ∎

Corollary 3.6**.**

Let $r\in{\mathbb{N}}$ and $p>4$ be a prime. We have

[TABLE]

Proof.

It follows from [10, Theorem 1], [14, Theorem 1.1], [16, Main Theorem], and [9, Theorem 2] that

[TABLE]

So Theorem 3.4 yields the corollary immediately. ∎

In fact, this note was motivated by Shen and Cai’s proof of (8) and a finer version of (9) in [7]. Now it follows from [10, Theorem 4] and [5, Theorem 1.1] that

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and, by similar computation (see [2] for details)

[TABLE]

Therefore, by Theorem 3.4, modulo $p^{r}$ ($r\geq 2$), we have

[TABLE]

By combining Theorem 3.4 and the numerical results of $S_{n}^{(1)}(p^{2})$ obtained in [2], one can derive easily similar explicit formulas for all $n\leq 12$.