The Hankel power sum matrix inverse and the Bernoulli continued fraction

Abstract

The m × m m \times m Hankel power sum matrix W = V V T W = V{V^T} (where V is the m × n m \times n Vandermonde matrix) has (i, j)-entry S i + j − 2 ( n ) {S_{i + j - 2}}(n) , where S p ( n ) = Σ k = 1 n k p {S_p}(n) = \Sigma _{k = 1}^n{k^p} . In solving a statistical problem on curve fitting it was required to determine f ( m ) f(m) so that for n > f ( m ) n > f(m) all eigenvalues of W − 1 {W^{ - 1}} would be less than 1. It is proved, after calcu lating W − 1 {W^{ - 1}} by first factoring W into easily invertible factors, that f ( m ) = ( 13 m 2 − 5 ) / 8 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 {m^{ - 1}} + {(m + 1)^{ - 1}} is found for the divergent Bernoulli number series Σ B 2 k ( 2 x ) 2 k \Sigma {B_{2k}}{(2x)^{2k}} .