Congruences involving generalized Catalan numbers and Bernoulli numbers

1.   Introduction

In combinatorics,

are the well-known Catalan numbers. The meaning of Catalan numbers are the numbers of ways to divide the $(n+2)$-polygon in $n$ triangles. For any positive integer $n$, the generalized Catalan numbers $B_{n, k}$ are defined (cf.^[10,15]) by

In ^[15], L. W. Shapiro shows that the meaning of the generalized Catalan numbers $B_{n, k}$ are the number of pairs of non-intersecting paths of length $n$ and distance $k$. For $1\leq k\leq n$, we list the first values of generalized Catalan numbers in the following table:

The generalized Catalan numbers satisfy the recurrence relation

with the initial conditions $B_{n, 0} = B_{n, m} = 0, \ m > n$. When $k = 1$, we have $B_{n, 1} = C_n$ for $n\geq1$.

Now we consider the generating function of the generalized Catalan numbers. Let

Exchanging the order of summation, we can get

In view of [4,(1.121)], we have

Combining (1.2) and (1.3) yields that

Remark. Taking $k = \frac{1}{2}$ in (1.3), we have

which implies that $z$ is the generating function of the Catalan numbers $C_n$.

There are various identities and congruences involving Catalan numbers (cf.^[5,6,11]). Differential equations and generating function are often used to manage combinatorial identities involving Catalan numbers (cf.^[8,9]). However, there are few identities involving the numbers $B_{n, k}$. Several applications of $B_{n, k}$ appeared in ^[1,6,15]. Koparal and Ömür ^[2,10,14] studied the congruences involving $B_{p, k}$, where $p$ is prime.

The numbers $B_{p, k}$ are closely related to generalized harmonic numbers under congruence relation. For $\alpha\in \mathbb{N}$, the generalized harmonic numbers are defined by

By the well-known Wolstenholme theorem^[20], we have that if $p > 3$ is a prime, then

For $m\in\{-2, -1, 0, 1, 2, 3\}$ and $n\in\{1, 2, 3\}$, Z.-W. Sun ^[16] established a kind of congruences $\mod p$ involving the sums $\sum_{k = 1}^{p-1}k^mH_k^n$. Y. Wang ^[18,19] generalized some of these congruences to $\mod p^2$ type. In ^[16], Z. W. Sun also made two conjectures on supercongruences of Euler-type. These conjectures were conformed in ^[13,17], respectively.

In this paper, we focus on the properties of $B_{n, k}$. With the use of the congruences involving harmonic numbers, we establish several congruences $\mod p^3$ involving the sums $\sum_{k = 1}^{p-1}k^mB_{p, k}^{2l}$ and $\mod p^2$ involving the sums $\sum_{k = 1}^{p-1}k^mB_{p, k}^{2l_1}B_{p, k-d}^{2l_2}$.

Our main results are as follows.

Theorem 1.1. Let $p > 3$ be a prime and $m, l$ be two positive integers such that $3\leq m < p-1$. Then

where $B_0, B_1, B_2, \ldots$ are the Bernoulli numbers defined by

Corollary 1.2. Let $p > 3$ be a prime and $m$ be an integer such that $3 < m < p-1$. Then

Corollary 1.3. Let $p > 3$ be a prime and $m$ be an even integer such that $3 < m < p-1$. If $p\mid l$ or $p\mid (4l-1)$ or $m = p-3$, then

Example 1.4. Let $p > 5$ be a prime and $m$ be an even integer such that $3 < m < p-1$. If $4\mid (\lambda p+1)$, then

In particular, for $\lambda = 3p$, we have

For a fixed positive integer $m$, we can use Theorem 1.1 to calculate the corresponding congruence. When $m$ is related to $p$, in general, we can not give a closed form. With the use of the known congruence, we give the following corollary.

Corollary 1.5. Let $p > 7$ be a prime and $l$ be a positive integer. Then

Now, we extend the definition of the generalized Catalan numbers by setting

From this, we see that

In this case, the generalized Catalan numbers satisfy the recurrence relation

with the initial conditions $B_{n, 0} = B_{n, m} = 0, \ |m| > n$.

Theorem 1.6. Let $p > 3$ be a prime and $m, l$ be two positive integers such that $1\leq m < p-3$. Then

Corollary 1.7. Let $p > 3$ be a prime and $m, l$ be two positive integers such that $m\in\{1, 3, \ldots, p-4\}$. If $l\equiv \frac{m(m+1)}{2(m-1)(m+2)}\pmod p$, then

Theorem 1.8. Let $p > 5$ be a prime and $m, l_1, l_2, d$ be positive integers such that $d$ is less than $p-1$ and $2\leq m < p-3$. Then

where $B_n(x) = \sum_{k = 0}^n{n\choose k }B_kx^{n-k}\ (n = 0, 1, 2, \ldots)$ are the Bernoulli polynomials.

Corollary 1.9. Let $p > 5$ be a prime and $m, l_1, l_2, d$ be positive integers such that $d$ is less than $p-1$ and $2\leq m < p-3$. Then

Corollary 1.10. Let $p > 5$ be a prime and $m > 1$ be an odd integer and $d$ be a positive integer less than $p-1$. Then

In particular, for $d = 1$, we have

Corollary 1.11. Let $p > 5$ be a prime and $m, l_1, l_2, d$ be positive integers such that $d$ is less than $p-1$ and $2\leq m < p-3$. Then

In particular, for $l_2 = p-1-l_1$, we have

In the next section, we provide some lemmas. In section $3$, we show the proof of the main results.

2.   Preliminaries

In this section, we first state some basic facts which will be used very often.

Lemma 2.1. Let $p > 3$ be a prime. If $m\in\{1, 2, \ldots, p-2\}$, then

In particular, for $m\in\{1, 3, 5, \ldots, p-4\}$, we have

Proof. These two congruences are due to J. W. L. ^[3]. □

Lemma 2.2. Let $p > 3$ be a prime and $k$ be an integer such that $1\leq k\leq p-1$. Then

Proof. According to the definition of generalized Catalan numbers, it follows that

Observe that

From this it is not difficult to deduce that

By the definition of generalized harmonic numbers, we have

It follows from (2.1) that

Combining (2.4) and (2.5) gives that

Hence

A simple calculation gives (2.3). □

Lemma 2.3. Let $p > 3$ be a prime and $m$ be a non-negative integer less than $p-1$. Then

For $m\geq3$, we have

Proof. Sun [16,(2.3)] showed that $\sum_{k = 1}^{p-1}H_k\equiv 1-p \pmod{p^3}$ and Wang [18,Theorem3.1] proved the rest cases of (2.8). The congruence (2.9) is the special case of the result of Wang [18,Theorem3.2]. □

Lemma 2.4. Let $p > 5$ be a prime. Then

For $p > 7$, we have

Proof. For any integer $m$ and $r$, we have the congruence

Suppose $m$ is an even integer. Then we have $(-1)^rB_rB_{p-1-m-r} = B_rB_{p-1-m-r}$ for $p > m+3$. Therefore,

Taking $m = 2$ in (2.12), we have

Observe that

which implies that

Zhao[21,(3.19)] showed that

Substituting (2.14) and (2.15) into (2.13) gives (2.10).

Taking $m = 4$ in (2.12), we have

Matiyasevich^[12] proved that for an even integer $n\geq4$, we have

Taking $n = p-5$ in the above identity, we can obtain

Zhao [21,Proposition 3.13] proved that

For any positive integer $i$, we have

Combining (2.16) through (2.19) gives (2.11). □

Lemma 2.5. Let $p$ be an odd prime. Suppose $s$ and $t$ are two positive integers of same parity such that $p > s+t+1$. Then

Proof. The congruence (2.20) is proved by Zhao[21,Theorem 3.2]. □

3.   Proof of the main results

Proof of Theorem 1.1. It follows from (2.3) that

which implies that

Hence

It is well-known that for $m, p\in\mathbb{Z}^+$,

(cf.[7,pp.230–238].) Hence for $3\leq m < p-1$, the congruence (3.2) reduces to

Suppose $m$ is an odd integer. It follows from (2.8) and (2.9) that

For the odd integer $m$, the integers $r$ and $m-r$ are of different parity. Recall that $B_1 = -1/2$ and $B_i = 0$ if $i\geq3$ and $2\nmid i$. Therefore,

Together with (2.8) and (3.6), we obtain

Substituting (3.5) and (3.7) into (3.4) gives the first congruence of Theorem 1.1.

If $m$ is an even integer such that $4\leq m < p-1$, then $\sum_{k = 1}^{p-1}k^{m-1}H_k\equiv 0\pmod p$. Substituting (2.8) and the second congruence of (2.9) into (3.4) gives the second congruence of Theorem 1.1. The proof of Theorem 1.1 is completed. □

Proof of Corollary 1.2. It follows from Theorem 1.1. □

Proof of Corollary 1.3. Case $p\mid l$ or $p\mid 4l-1$ is obvious by the second congruence of Theorem 1.1. Case $m = p-3$ follows from (2.10). □

Proof of Corollary 1.5. Combining (2.11) and the second congruence of Theorem 1.1 gives the desired congruence. □

Proof of Theorem 1.6. In view of (3.1), we have

Suppose $m$ is an odd integer. It follows from (2.1) that $H_{p-1}^{(m+2)}\equiv 0\pmod p$ and

By Fermat's little theorem, we have $\sum_{k = 1}^{p-1}H_k/k^{m+1}\equiv \sum_{k = 1}^{p-1}k^{p-2-m}H_k \pmod p$. In view of (2.8), we have $\sum_{k = 1}^{p-1}H_k/k^{m+1}\equiv B_{p-2-m}\pmod p$. Similarly, we can obtain $\sum_{k = 1}^{p-1}H_k/k^{m}\equiv B_{p-1-m}\equiv0\pmod p$, since $m$ is odd and $1\leq m \leq p-4$. Also, we can get $\sum_{k = 1}^{p-1}H_k^2/k^{m}\equiv B_{p-2-m}\pmod p$ by (2.9). Combining the above and (2.2), we obtain

Observe that

Hence taking $s = 1, t = m$ in (2.20), we have

Substituting (3.11) into (3.9), we get the first congruence.

Now we assume that $m$ is an even integer. With the use of (2.8), we have $\sum_{k = 1}^{p-1}H_k/k^{m}\equiv B_{p-1-m}\pmod p$. Combining (2.1) and (3.8), we obtain the second congruence. □

Proof of Corollary 1.7. It follows from Theorem 1.6. □

Proof of Theorem 1.8. According to the definition of the generalized Catalan numbers, it is easy to see that $B_{p, 0} = 0$ and

With the use of (1.5) and (2.3), we can obtain the following congruences. If $k > d$, then

If $k < d$, then

Combining (3.12) through (3.14), it follows that

The above congruence can be written as

where $\Sigma_1 = \sum_{k = d+1}^{p-1}k^m \left(\frac{1}{k-d}-2H_{k-d}\right)$ and $\Sigma_2 = \sum_{k = 1}^{d-1}k^m \left(\frac{1}{d-k}-2H_{d-k}\right)$. Observe that

Recall that $H_{p-d+k}\equiv H_{d-k-1}\pmod p$ for $k < d$. Then

which implies that

With the use of the binomial theorem, we have

Exchanging the summation order gives that

For $m\in\mathbb{Z}$, there holds

(cf.[7,pp.235].) Hence

In view of (2.8) and the definition of Bernoulli polynomials, we have

Writing (3.17) and (3.18) into (3.16), we obtain

In view of (3.3), we have $\sum_{k = 1}^{p-1}k^m\equiv p B_m\pmod {p^2}$ and $\sum_{k = 1}^{p-1}k^{m-1}\equiv 0\pmod {p}$ for $m > 2$. Hence with the help of (2.8), we have

Substituting (3.19) and (3.20) into (3.15), we complete the proof of the theorem. □

Proof of Corollary 1.9 and 1.10. These congruences can be obtained directly from Theorem 1.8. □

Proof of Corollary 1.11. Replacing $l_1, l_2$ by $l_1p, l_2p$ in Theorem 1.8 respectively, we get the first congruence. Then taking $l_2 = p-1-l_1$ and with the use of Euler theorem, we obtain the second one. □

4.   Conclusions

Koparal and Ömür ^[10] established a kind of mod $p^2$ congruences involving the sum $\sum_{k = 1}^{p-1}k^2B_{p, k}B_{p, k-d}$ and $\sum_{k = 1}^{p-1}\frac{B_{p, k}B_{p, k-d}}{k}$, where $B_{p, k}$ is generalized Catalan numbers and the power of $B_{p, k}$ is odd. This article studies the congruences involving generalized Catalan numbers with even powers and establishes a kind of mod $p^3$ congruences involving the sum $\sum_{k = 1}^{p-1}k^mB_{p, k}^{2l}$ and mod $p^2$ congruences involving the sum $\sum_{k = 1}^{p-1}k^mB_{p, k}^{2l_1}B_{p, k-d}^{2l_2}$, where $m$ is an integer and $l, l_1, l_2$ are positive integers.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgements

We would like to thank the anonymous referees for many valuable suggestions. This research was supported by Natural Science Foundation of China (Grant No. 11871258, 12271234) and supported by the Young backbone teachers in Henan Province (Grant No. 2020GGJS194) and supported by the Young backbone teachers in Luoyang Normal College (Grant No. 2019XJGGJS-10).

Conflict of interest

The authors declare no conflicts of interest.