A ug 2 00 7 Determinants related to binomial coefficients modulo 2 and 4

The main results of this paper are computations of a few determinants related to binomial coefficients. The most interesting example, discovered after browsing through [5], is obtained by considering binomial coefficients modulo 4. We include related results contained in [2] and [3]. The next section recalls mostly well-known facts concerning binomial and q−binomial coefficients and states the main results. It contains also a (new?) formula for evaluating q−binomials coefficients at roots of unity. Section 3 describes the algebra of recurrence matrices which is a convenient tool for proofs. Section 4 proves formulae for the determinant of the reduction modulo 2 of the symmetric Pascal matrix (already contained in [2]) and of a determinant related to the 2−valuation of the binomial coefficients (essentially contained in [3]). Section 5 is devoted to the proof of the main result, a determinant associated to the “Beeblebrox reduction” (defined as β(n) = 0 if n ∈ 2Z and β(n) = ±1 ≡ n (mod 4) for odd n) of binomial coefficients. It contains also a digression on the “lower triangular Beeblebrox matrix” and an associated group. Section 6 contains a proof of a (new?) formula for evaluating q−binomial coefficients at roots of unity. This formula yields easily some determinants associated to the reduction modulo 2 and the Beeblebrox reduction of (real and imaginary parts) of q−binomial coefficients evaluated at q = −1 and q = i.