Supercongruences involving products of three binomial coefficients

Let p > 3 be a prime, and let a be a rational p -adic integer. Using the WZ method we establish the congruences for ∑ k = 0 p - 1 a k - 1 - a k 2 k k w ( k ) 4 k modulo p 3 , where w ( k ) ∈ { 1 , 1 k + 1 , 1 ( k + 1 ) 2 , 1 2 k - 1 } . Taking a = - 1 2 , - 1 3 , - 1 4 , - 1 6 in the congruences confirms some conjectures posed by the author earlier.