Universal Peculiar Linear Mean Relationships in All Polynomials

Abstract

In any cubic polynomial, the average of the slopes at the 3 roots is the negation of the slope at the average of the roots. In any quartic, the average of the slopes at the 4 roots is twice the negation of the slope at the average of the roots. We generalize such situations and present a procedure for determining all such relationships for polynomials of any degree. E.g., in any septic f , letting f n denote the mean f value over all zeroes of the derivative f (n) , it holds that 37 f 1 − 150 f 3 + 200 f 4 − 135 f 5 + 48 f 6 = 0; and in any quartic it holds that 5 f 1 − 6 f 2 + 1 f 3 = 0. Having calculated such relationships in all dimensions up to 49, in all even dimensions there is a single relationship, in all odd dimensions there is a twodimensional family of relationships. We see connections to Tchebyshev, Bernoulli, & Euler polynomials, and Stirling numbers.