## The Hankel power sum matrix inverse and the Bernoulli continued fraction

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- by J. S. Frame PDF
- Math. Comp.
**33**(1979), 815-826 Request permission

## 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

*m*th partial denominator ${m^{ - 1}} + {(m + 1)^{ - 1}}$ is found for the divergent Bernoulli number series $\Sigma {B_{2k}}{(2x)^{2k}}$.

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## Additional Information

- © Copyright 1979 American Mathematical Society
- Journal: Math. Comp.
**33**(1979), 815-826 - MSC: Primary 65F30
- DOI: https://doi.org/10.1090/S0025-5718-1979-0521297-0
- MathSciNet review: 521297