A combinatorial determinant

A theorem of Mina evaluates the determinant of a matrix with entries D(f(x)). We note the important special case where the matrix entries are evaluated at x = 0 and give a simple proof of it, as well as some special additivity properties that hold in this case, but not in general. Some applications are given also. We then give a short proof of the general case. October 25, 1998 An old theorem of Mina [1], which “deserves to be better known” [3], states that det  ( dj dxj f(x) )n−1 i,j=0  = 1!2! . . . (n− 1)!f ′(x)n(n−1)/2. (1) A proof of Mina’s theorem can be found, for instance, in [2]. We will first give a short proof of the special case in which both sides are evaluated at x = 0, with some applications, and then give a short proof of the general case. The special case shows an interesting structure owing to the fact that all matrices of that form can be simultaneously triangularized by multiplying them by a certain universal triangular matrix. 1 Coefficients of powers of power series Theorem 1 Let f = 1 + a1x+ a2x + . . . be a formal power series, and define a matrix c by1 ci,j = [x]f i (i, j ≥ 0). (2) Then det ( (ci,j)i,j=0 ) = a 1 (n = 0, 1, 2, . . .). (3) 1 “[x]g” means the coefficient of x in the series g. 1 To prove this, define a matrix b by bi,j = (−1) ( i j ) , (i, j ≥ 0). (4) Then we claim that bc is upper triangular with powers of a1 on its diagonal. Indeed we have ∑