On Wendt’s determinant

Abstract

Wendt’s determinant of order m m is the circulant determinant W m W_{m} whose ( i , j ) (i,j) -th entry is the binomial coefficient ( m | i − j | ) \binom m{|i-j|} , for 1 ≤ i , j ≤ m 1\leq i,j\leq m . We give a formula for W m W_{m} , when m m is even not divisible by 6, in terms of the discriminant of a polynomial T m + 1 T_{m+1} , with rational coefficients, associated to ( X + 1 ) m + 1 − X m + 1 − 1 (X+1)^{m+1}-X^{m+1}-1 . In particular, when m = p − 1 m=p-1 where p p is a prime ≡ − 1 ( m o d 6 ) \equiv -1 (mod 6) , this yields a factorization of W p − 1 W_{p-1} involving a Fermat quotient, a power of p p and the 6-th power of an integer.