Proof of Sun's conjectural supercongruence involving Catalan numbers

We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.


1.
Introduction. In 2003, Rodriguez-Villegas [14] conjectured the following four supercongruences associated to certain elliptic curves: where p ≥ 5 is a prime and · p denotes the Legendre symbol. These four supercongruences were first proved by Mortenson [12,13] by using the Gross-Koblitz formula. Guo, Pan and Zhang [3] established some interesting q-analogues of the above four supercongruences. For more q-analogues of congruences, one can refer to [1,2,4,5,10].
Recall that the Euler numbers are defined as and the nth Catalan number is given by which plays an important role in various counting problems. We refer to [17] for many different combinatorial interpretations of the Catalan numbers.
Conjecture 1.1 (Sun, 2019). For any prime p ≥ 5, we have The main purpose of the paper is to prove (2). Our proof is based on hypergeometric series identities and symbolic summation method.
We establish two preliminary results in the next section. The proof of Theorem 1.2 will be given in Section 3.
2. Preliminary results. In order to prove Theorem 1.2, we need the following two key results.
where the Bernoulli numbers are given by Before proving Proposition 2.1, we establish the following lemma.

It follows that
Letting x → 1 on both sides of (10) and noting that , we arrive at which proves (6).

.2.2), page 65])
Letting d = −n, f = 1 2 and g = −n + 1 2 in (24), we obtain which is (22). On the other hand, (23) can be discovered and proved by symbolic summation package Sigma due to Schneider [15]. One can refer to [9] for the same approach to finding and proving identities of this type.