PRODUCTS OF BINOMIAL COEFFICIENTS MODULO p 2

As usual Z, Q, R and C denote the ring of integers, the rational field, the real field and the complex field respectively. We also let Z = {1, 2, 3, · · · } and C∗ = C \ {0}. For a ∈ Z and n ∈ Z, by (a, n) we mean th greatest common divisor of a and n, if n is odd then the Jacobi symbol ( a n ) is defined in terms of Legendre symbols (see, e.g. [IR]). For x ∈ R, [x] and {x} stand for the integral and the fractional parts of x respectively. For a prime p and an integer a prime to p, the Fermat quotient (ap−1 − 1)/p is denoted by qp(a). For an odd prime p and a ∈ Z, we define the Euler quotient