OFFSET

1,1

COMMENTS

Irregular triangular array of coefficients of universal linear relationships among means of all even-degree (complex-valued) polynomials, beginning with degree = 4 at top of triangle. Let phi(D,d,r) denote the mean of a generic degree D polynomial's order d derivative averaged over the (D-r) roots of the order r derivative of the polynomial. The tabulated coefficients, c(-), satisfy Sum_{k=1..(D-d-1)} c(k)*phi(D,d,k) = 0, with d always equal to 0. The first row of the triangle has 3 entries, while each subsequent row has an additional 2 entries.

Results have been computed in all degrees up to D=49, observing: (1) in all even degrees beyond 2, there is a single such linear relationship; (2) in all odd degrees beyond 3, there is a 2-dimensional family of such linear relationships.

In each row of the triangle, the sum of all positive coefficients equals the sum of all negative coefficients.

LINKS

G. G. Wojnar, D. Sz. Wojnar, and L. Q. Brin, Universal Peculiar Linear Mean Relationships in All Polynomials, arXiv:1706.08381 [math.GM], 2017.

EXAMPLE

Triangle begins:

5, -6, 1;

77, -120, 60, -20, 3;

669, -1260, 1050, -700, 315, -84, 10;

4609, -10080, 11760, -11760, 8820, -4704, 1680, -360, 35;

...

Example 1: For any polynomial of degree D=4, it holds that 5*phi(4,0,1) - 6*phi(4,0,2) + 1*phi(4,0,3) = 0.

Example 2: For any polynomial of degree D=6, it holds that 77*phi(6,0,1) - 120*phi(6,0,2) + 60*phi(6,0,3) - 20*phi(6,0,4) + 3*phi(6,0,5) = 0.

CROSSREFS

KEYWORD

sign,tabf

AUTHOR

Gregory Gerard Wojnar, Sep 30 2017

STATUS

approved