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The Hankel power sum matrix inverse and the Bernoulli continued fraction

Author: J. S. Frame
Journal: Math. Comp. 33 (1979), 815-826
MSC: Primary 65F30
MathSciNet review: 521297
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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}}$.

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Article copyright: © Copyright 1979 American Mathematical Society

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