Polynomials generalizing binomial coefficients and their application to the study of Fermat's last theorem

Abstract

We study properties of the polynomials φ k ( X) which appear in the formal development Π k − 0 n ( a + bX k ) rk = Σ k ≥ 0 φ k ( X) a r − k b k , where r k ∈ l and r = Σ r k . this permits us to obtain the coefficients of all cyclotomic polynomials. Then we use these properties to expand the cyclotomic numbers G r ( ξ) = Π k = 1 p − 1 ( a + bξ k ) k r , where p is a prime, ξ is a primitive pth root of 1, a, b ∈ l and 1 ≤ r ≤ p − 3, modulo powers of ξ − 1 (until ( ξ − 1) 2( p − 1) − r ). This gives more information than the usual logarithmic derivative. Suppose that p ∤ ab(a + b). Let m = − b a . We prove that G r ( ξ) ≡ c p mod p( ξ − 1) 2 for some c ∈ l , if and only if Σ k = 1 p − 1 k p − 2 − r m k ≡ 0 (mod p). We hope to show in this work that this result is useful in the study of the first case of Fermat's last theorem.