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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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  • [1] W. A. Al-Salam and L. Carlitz, Some determinants of Bernoulli, Euler and related numbers, Portugal. Math. 18 (1959), 91–99. MR 0123523
  • [2] J. S. Frame, The solution of equations by coninued fractions, Amer. Math. Monthly 60 (1953), 293–305. MR 0056369
  • [3] J. S. Frame, Bernoulli numbers modulo 27000, Amer. Math. Monthly 68 (1961), 87–95. MR 0124272,
  • [4] D. C. GILLILAND & JAMES HANNAN, Detection of Singularities in the Countable General Linear Model, Department of Statistics, Michigan State University, RM-217, DCG-8, JH-10, Aug. 1971.
  • [5] Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039
  • [6] N. E. NÖRLUND, Vorlesung über Differenzenrechnung, Springer, Berlin, 1924, p. 18.
  • [7] G. M. Phillips and P. J. Taylor, Theory and applications of numerical analysis, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1973. MR 0343523
  • [8] T. J. STIELTJES, "Sur quelques intégrales définies et leur dévéloppement en fractions continues," Oeuvres Complètes, vol. 2, P. Noordhoff, Groningen, 1918, pp. 378-391.
  • [9] H. S. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948. MR 0025596

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Article copyright: © Copyright 1979 American Mathematical Society