Two congruences concerning Ap\'{e}ry numbers

Let $n$ be a nonnegative integer. The $n$-th Ap\'{e}ry number is defined by $$ A_n:=\sum_{k=0}^n\binom{n+k}{k}^2\binom{n}{k}^2. $$ Z.-W. Sun ever investigated the congruence properties of Ap\'{e}ry numbers and posed some conjectures. For example, Sun conjectured that for any prime $p\geq7$ $$ \sum_{k=0}^{p-1}(2k+1)A_k\equiv p-\frac{7}{2}p^2H_{p-1}\pmod{p^6} $$ and for any prime $p\geq5$ $$ \sum_{k=0}^{p-1}(2k+1)^3A_k\equiv p^3+4p^4H_{p-1}+\frac{6}{5}p^8B_{p-5}\pmod{p^9}, $$ where $H_n=\sum_{k=1}^n1/k$ denotes the $n$-th harmonic number and $B_0,B_1,\ldots$ are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.

Z.-W. Sun ever investigated the congruence properties of Apéry numbers and posed some conjectures. For example, Sun conjectured that for any prime p ≥ 7 and for any prime p ≥ 5 where H n = n k=1 1/k denotes the n-th harmonic number and B 0 , B 1 , . . . are the well-known Bernoulli numbers. In this paper we shall confirm these two conjectures.

Introduction
The well-known Apéry numbers given by were first introduced by Apéry to prove the irrationality of ζ(3) = ∞ n=1 1/n 3 (see [1,6]). In 2012, Z.-W. Sun introduced the Apéry polynomials where (−) denotes the Legendre symbol. Letting x = 1 and for any prime p ≥ 5, Sun established the following generalization of (1.1): where B 0 , B 1 , . . . are the well-known Bernoulli numbers defined as follows: In 1850 Kummer (cf. [4]) proved that for any odd prime p and any even n . From [3] we know that H p−1 ≡ −p 2 B p−3 /3 (mod p 3 ) for any prime p ≥ 5. Thus (1.2) has the following equivalent form Motivated by Sun's work on Apéry polynomials, V.J.W. Guo and J. Zeng studied the divisibility of the following sums: Particularly, for r = 1, they obtained where p ≥ 5 is a prime. As an extension to (1.7), Sun [8, Conjecture A65] proposed the following challenging conjecture.
Conjecture 1.1. For any prime p ≥ 5 we have This is our second theorem.
Proofs of Theorems 1.1-1.2 will be given in Sections 2-3 respectively.

Proof of Theorem 1.1
The proofs in this paper strongly depend on the congruence properties of harmonic numbers and the Bernoulli numbers. (The readers may consult [4,7,9,11] for the properties of them.) Below we first list some congruences involving harmonic numbers and the Bernoulli numbers which may be used later.
and H (2) respectively. By Lemma 2.1, we immediately obtain that H p−1 ≡ −pH (2) p−1 /2 (mod p 4 ). Thus Recall that the Bernoulli polynomials B n (x) are defined as for any positive integer n and m. Let d > 0 and s := (s 1 , . . . , s d ) ∈ (Z\{0}) d . The alternating multiple harmonic sum [12] is defined as follows Let A, B, D, E, F be defined as in [12,Section 6], i.e., Proof. In [12, Section 6], Tauraso and Zhao proved that Combining the above two congruences we immediately obtain the desired result.
Lemma 2.4. Let p ≥ 7 be a prime. Then we have Proof. By Lemma 2.2, it is easy to check that where the last step follows from the fact B n = 0 for any odd n ≥ 3. By [4] we know that B n (1/2) = (2 1−n − 1)B n . Thus With helps of Lemmas 2.2 and 2.3, we have Combining this with (2.5), we have completed the proof of Lemma 2.4.

Proof of Theorem 1.2
In order to show Theorem 1.2, we need the following results.
Lemma 3.1. Let p ≥ 7 be a prime. Then we have Proof. By Remark 2.1 we have On one hand, On the other hand, we have From the above and with the help of Lemma 3.2, we obtain (3.3). Now we turn to prove (3.4). It is easy to see that By [12, The proof of Lemma 3.3 is now complete.
Lemma 3.4. Let k ∈ N. Then for n ∈ Z + we have Proof. It can be verified directly by induction on n.