Generalization of Wolstenholme's and Morley's congruences

Super-congruences involving trininomial coefficients

“…In 1895, Morley [15] proved, for any prime number p ≥ 5, that p − 1 (p − 1) /2 ≡ (−1) (p−1)/2 4 p−1 = (−1) (p−1)/2 (1 + pq 2 ) 2 mod p 3 , where q a is the Fermat quotient defined for a given prime number p by q a = q a (p) := a p−1 − 1 p , a ∈ Z−pZ, and Z denotes the set of the integer numbers. Also, in 1953, Carlitz [6,7] improved, for any prime number p ≥ 5, Morley's congruence to (−1)…”

“…Many great mathematicians have been interested to generalize the congruence of Wostenhlom and Morly, such the works of Zhao [19], McIntosh [13], Meštrović [14], Bencherif et al [3] and Sun [16]. Recently, Sun [17] gave some properties and congruences involving the coefficients n n 2 defined by…”

“…We note here that (−1) [(p−1)/3] = 1 if p ≡ 1 (mod 3) and −1 if p ≡ 2 (mod 3) . Hence, by combining this congruence with the congruence (1), we obtain the congruence (3). Set L = np − 1, m = (p − 1) /2 in (28).…”