The Hankel power sum matrix inverse and the Bernoulli continued fraction
Author:
J. S. Frame
Journal:
Math. Comp. 33 (1979), 815826
MSC:
Primary 65F30
MathSciNet review:
521297
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The Hankel power sum matrix (where V is the Vandermonde matrix) has (i, j)entry , where . In solving a statistical problem on curve fitting it was required to determine so that for all eigenvalues of would be less than 1. It is proved, after calcu lating by first factoring W into easily invertible factors, that suffices. As byproducts of the proof, close approximations are given for the Hilbert determinant, and a convergent continued fraction with mth partial denominator is found for the divergent Bernoulli number series .
 [1]
W.
A. AlSalam and L.
Carlitz, Some determinants of Bernoulli, Euler and related
numbers, Portugal. Math. 18 (1959), 91–99. MR 0123523
(23 #A848)
 [2]
J.
S. Frame, The solution of equations by coninued fractions,
Amer. Math. Monthly 60 (1953), 293–305. MR 0056369
(15,65b)
 [3]
J.
S. Frame, Bernoulli numbers modulo 27000, Amer. Math. Monthly
68 (1961), 87–95. MR 0124272
(23 #A1586)
 [4]
D. C. GILLILAND & JAMES HANNAN, Detection of Singularities in the Countable General Linear Model, Department of Statistics, Michigan State University, RM217, DCG8, JH10, Aug. 1971.
 [5]
Eugene
Isaacson and Herbert
Bishop Keller, Analysis of numerical methods, John Wiley &
Sons Inc., New York, 1966. MR 0201039
(34 #924)
 [6]
N. E. NÖRLUND, Vorlesung über Differenzenrechnung, Springer, Berlin, 1924, p. 18.
 [7]
G.
M. Phillips and P
Taylor J., Theory and applications of numerical analysis,
Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers],
LondonNew York, 1973. MR 0343523
(49 #8264)
 [8]
T. J. STIELTJES, "Sur quelques intégrales définies et leur dévéloppement en fractions continues," Oeuvres Complètes, vol. 2, P. Noordhoff, Groningen, 1918, pp. 378391.
 [9]
H.
S. Wall, Analytic Theory of Continued Fractions, D. Van
Nostrand Company, Inc., New York, N. Y., 1948. MR 0025596
(10,32d)
 [1]
 W. A. ALSALAM & L. CARLITZ, "Some determinants of Bernoulli, Euler, and related numbers," Portugal. Math., v. 18, 1959, pp. 9199. MR 0123523 (23:A848)
 [2]
 J. S. FRAME, "The solution of equations by continued fractions," Amer. Math. Monthly, v. 60, 1953, pp. 293305. MR 0056369 (15:65b)
 [3]
 J. S. FRAME, "Bernoulli numbers modulo 27000," Amer. Math. Monthly, v. 68, 1961, pp. 8795. MR 0124272 (23:A1586)
 [4]
 D. C. GILLILAND & JAMES HANNAN, Detection of Singularities in the Countable General Linear Model, Department of Statistics, Michigan State University, RM217, DCG8, JH10, Aug. 1971.
 [5]
 E. ISAACSON & H. B. KELLER, Analysis of Numerical Methods, Wiley, New York, 1966, pp. 196, 217218. MR 0201039 (34:924)
 [6]
 N. E. NÖRLUND, Vorlesung über Differenzenrechnung, Springer, Berlin, 1924, p. 18.
 [7]
 G. M. PHILLIPS & P. J. TAYLOR, Theory and Application of Numerical Analysis, Academic Press, New York, 1973, pp. 91, 246. MR 0343523 (49:8264)
 [8]
 T. J. STIELTJES, "Sur quelques intégrales définies et leur dévéloppement en fractions continues," Oeuvres Complètes, vol. 2, P. Noordhoff, Groningen, 1918, pp. 378391.
 [9]
 H. S. WALL, Analytic Theory of Continued Fractions, D. Van Nostrand, Princeton, N. J., 1948, pp. 369376. MR 0025596 (10:32d)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65F30
Retrieve articles in all journals
with MSC:
65F30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905212970
PII:
S 00255718(1979)05212970
Article copyright:
© Copyright 1979 American Mathematical Society
