On two supercongruences for sums of Apéry-like numbers (Q2050223)

The author proved that for any prime \(p>3\), we have \[ \sum_{k=0}^{p-1}(2k+1)\frac{T_k}{4^k}\equiv p+\frac76p^4B_{p-3}\pmod{p^5} \] and \[ \sum_{k=0}^{p-1}(2k+1)\frac{T_k}{(-4)^k}\equiv(-1)^{(p-1)/2}p+p^3E_{p-3}\pmod{p^4}. \] Those two congruences involving Apéry-like numbers were conjectured by Z.-H. Sun. The author proves the above two congruences by using some combinatorial identities involving harmonic numbers and some related congruences.