OFFSET
1,3
COMMENTS
Irregular triangular array of coefficients of universal linear relationships among means of all (complex-valued) polynomials, beginning with degree = 3 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.
Results have been computed in all degrees up to D=40, 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:
1, -1;
5, -6, 1;
1, -2, 2, -1;
2, -2, -1, 1;
77, -120, 60, -20, 3;
1, -3, 5, -5, 3, -1;
62, -75, -25, 75, -60, 23;
...
Example 1: For any polynomial of degree D=3, it holds that 1*phi(3,0,1) - 1*phi(3,0,2) = 0.
Example 2: 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 3: 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, May 27 2017
EXTENSIONS
More terms from Gregory Gerard Wojnar, Sep 24 2017
Edited by N. J. A. Sloane, Sep 30 2017
STATUS
approved