The Hankel power sum matrix inverse and the Bernoulli continued fraction

Abstract:

The $m \times m$ Hankel power sum matrix $W = V{V^T}$ (where V is the $m \times n$ Vandermonde matrix) has (i, j)-entry ${S_{i + j - 2}}(n)$, where ${S_p}(n) = \Sigma _{k = 1}^n{k^p}$. In solving a statistical problem on curve fitting it was required to determine $f(m)$ so that for $n > f(m)$ all eigenvalues of ${W^{ - 1}}$ would be less than 1. It is proved, after calcu lating ${W^{ - 1}}$ by first factoring W into easily invertible factors, that $f(m) = (13{m^2} - 5)/8$ suffices. As by-products of the proof, close approximations are given for the Hilbert determinant, and a convergent continued fraction with mth partial denominator ${m^{ - 1}} + {(m + 1)^{ - 1}}$ is found for the divergent Bernoulli number series $\Sigma {B_{2k}}{(2x)^{2k}}$.