Computation of Newton sum rules for associated and co-recursive classical orthogonal polynomials (Q5946621)

This is a review paper on known results on sums of powers of zeros of certain polynomials. NEWLINENEWLINENEWLINEGiven a polynomial NEWLINE\[NEWLINE P_N(x)=\prod_{i=1}^N (x-x_i)=x^N-u_1x^{N-1}+ \cdots +(-1)^Nu_N NEWLINE\]NEWLINE that satisfies the generalized hypergeometric differential equation NEWLINE\[NEWLINE \sum_{i=0}^r g_i(x)y^{(i)}(x)=0 NEWLINE\]NEWLINE where NEWLINE\[NEWLINE g_i(x)=\sum_{j=0}^I a_j^{(i)}x^j. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe authors then discuss a formula for the Newton sum rules NEWLINE\[NEWLINE y_h=\sum_{k=1}^N x_k^h NEWLINE\]NEWLINE and the Case sum rules NEWLINE\[NEWLINE J_h^{(i)}=\sum_{\not= (\ell_1,\ldots,\ell_i)}^{(1,\ldots,N)} {x_{\ell_1}^h\over \prod_{k=1}^i (x_{\ell_1}-x_{\ell_k})}, NEWLINE\]NEWLINE where the symbol \(\not=\) means that the sum is running over all \(\ell_j\;(j=1,\ldots,N)\), subject to the condition that all \(\ell\) are different.