Supercongruences involving binomial coefficients and Euler polynomials

Let p be an odd prime and let x be a p-adic integer. In this paper, we establish supercongruences for ∑ₖ₌₀ᵖ⁻¹xkx+kk(-4)ᵏ(dk+1)2kkp² and ∑ₖ₌₀ᵖ⁻¹xkx+kk(-2)ᵏ(dk+1)2kkp², where d∈\0,1,2\. As consequences, we extend some known results. For example, for $p>3$ we show ∑ₖ₌₀ᵖ⁻¹3kk(4/27)ᵏ≡19+89p+4/27pEₚ₋₂(13)p², where Eₙ(x) denotes the Euler polynomial of degree n. This generalizes a known congruence of Z.-W. Sun.