Gregory coefficientsGn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind,[1][2][3][4][5][6][7][8][9][10][11][12][13] are the rational numbers
that occur in the Maclaurin series expansion of the reciprocal logarithm
Gregory coefficients are alternating Gn = (−1)n−1|Gn| for n > 0
and decreasing in absolute value. These numbers are named after James Gregory who introduced them in 1670 in the numerical integration context. They were subsequently rediscovered by many mathematicians and often appear in works of modern authors, who do not always recognize them.[1][5][14][15][16][17]
^ abCh. Jordan. The Calculus of Finite Differences Chelsea Publishing Company, USA, 1947.
^L. Comtet. Advanced combinatorics (2nd Edn.) D. Reidel Publishing Company, Boston, USA, 1974.
^Cite error: The named reference davis was invoked but never defined (see the help page).
^P. C. Stamper. Table of Gregory coefficients. Math. Comp. vol. 20, p. 465, 1966.
^ abD. Merlini, R. Sprugnoli, M. C. Verri. The Cauchy numbers. Discrete Math., vol. 306, pp. 1906–1920, 2006.
^Cite error: The named reference nemes was invoked but never defined (see the help page).
^P.T. Young. A 2-adic formula for Bernoulli numbers of the second kind and for the Nörlund numbers. J. Number Theory, vol. 128, pp. 2951–2962, 2008.
^V. Kowalenko. Properties and Applications of the Reciprocal Logarithm Numbers. Acta Appl. Math., vol. 109, pp. 413–437, 2010.
^V. Kowalenko. Generalizing the reciprocal logarithm numbers by adapting the partition method for a power series expansion. Acta Appl. Math., vol. 106, pp. 369–420, 2009.
^Cite error: The named reference Alab1 was invoked but never defined (see the help page).
^Cite error: The named reference Alab2 was invoked but never defined (see the help page).
^F. Qi and X.-J. Zhang An integral representation, some inequalities, and complete monotonicity of Bernoulli numbers of the second kind. Bull. Korean Math. Soc., vol. 52, no. 3, pp. 987–98, 2015.
^Weisstein, Eric W. "Logarithmic Number." From MathWorld—A Wolfram Web Resource.
^Cite error: The named reference klvr was invoked but never defined (see the help page).
^J.F. Steffensen. Interpolation (2nd Edn.). Chelsea Publishing Company, New York, USA, 1950.
^Cite error: The named reference car was invoked but never defined (see the help page).
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and 24 Related for: Gregory coefficients information
Gregorycoefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the...
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are the signed Stirling numbers of the first kind and Gn are the Gregorycoefficients. The expansion of the Bernoulli polynomials of the second kind into...
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_{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}} whose coefficients are the generalized binomial coefficients ( α n ) = ∏ k = 1 n α − k + 1 k = α ( α − 1 ) ⋯...
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was popularised by David Hendry, who argued that lack of stability of coefficients frequently caused forecast failure, and therefore we must routinely test...
at equilibrium n ∗ {\displaystyle n^{*}} . The matrix of the kinetic coefficients L i j {\displaystyle L_{ij}} should be symmetric (Onsager reciprocal...
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respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin...
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