A note on summation formulas of powers of roots
Daniel C. Fielder · Mathematics of Computation · 1958
Introduction. Through application of a method due to Newton [1], it is possible to find solutions for the sum Sb of the bth powers of the n roots of f(x), where f(x) is an integral rational function of nth degree in x. As anyone who has used this method for large n and b can attest, the procedure involves tedious and repetitive manipulations which increase rapidly in complexity as b is increased. It is shown herein that it is possible to reduce the labor of finding such algebraic sums by rejecting redundant information in a general formula for the sums and using only pertinent terms to predict the numerical and literal coefficients. Inductive reasoning is employed to arrive at a simplified method. Conventional means for determining sums. Given the expression