Congruences involving generalized Catalan numbers and Bernoulli numbers

: In this paper, we establish some congruences mod p 3 involving the sums (cid:80) p − 1 k = 1 k m B 2 lp , k , where p > 3 is a prime number and B p , k are generalized Catalan numbers. We also establish some congruences mod p 2 involving the sums (cid:80) p − 1 k = 1 k m B 2 l 1 p , k B 2 l 2 p , k − d , where m , l 1 , l 2 , d are positive integers and 1 ≤ d ≤ p − 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 B n,k = k n 2n n − k , 0 ≤ k ≤ n.
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 ≤ k ≤ n, we list the first values of generalized Catalan numbers in the following table:  Remark.
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 andÖmür [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 α ∈ 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 ∈ {−2, −1, 0, 1, 2, 3} and n ∈ {1, 2, 3}, Z.-W. Sun [16] established a kind of congruences mod p involving the sums p−1 k=1 k m H n k . 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 p−1 k=1 k m B 2l p,k and mod p 2 involving the sums p−1 k=1 k m B 2l 1 p,k B 2l 2 p,k−d . 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 ≤ m < p − 1. Then   p,k ≡ (−1) In particular, for λ = 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 ≤ m < p − 3. Then Corollary 1.7. Let p > 3 be a prime and m, l be two positive integers such that m ∈ {1, 3, . . . , p − 4}.
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 ≤ m < p − 3. Then where B n (x) = n k=0 n k B k x n−k (n = 0, 1, 2, . . .) 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 ≤ 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 ≤ 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.

Preliminaries
In this section, we first state some basic facts which will be used very often.
In particular, for m ∈ {1, 3, 5, . . . , 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 ≤ k ≤ 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 Proof. For any integer m and r, we have the congruence Suppose m is an even integer. Then we have (−1) r B r B p−1−m−r = B r B p−1−m−r for p > m + 3. Therefore, Matiyasevich [12] proved that for an even integer n ≥ 4, we have Taking n = p − 5 in the above identity, we can obtain   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 (m+2) p−1 ≡ 0 (mod p) and Observe that With the use of (1.5) and (2.3), we can obtain the following congruences. If k > d, then (3.14) Combining (3.12) through (3.14), it follows that The above congruence can be written as Recall that H p−d+k ≡ H d−k−1 (mod p) for k < d. Then which implies that With the use of the binomial theorem, we have Exchanging the summation order gives that In view of (3.3), we have p−1 k=1 k m ≡ pB m (mod p 2 ) and p−1 k=1 k m−1 ≡ 0 (mod p) for m > 2. Hence with the help of (2.8), we have Proof of Corollary 1.11. Replacing l 1 , l 2 by l 1 p, l 2 p 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. , 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 p−1 k=1 k m B 2l p,k and mod p 2 congruences involving the sum p−1 k=1 k m B 2l 1 p,k B 2l 2 p,k−d , 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.