Numerical values of Goldberg’s coefficients in the series for 𝑙𝑜𝑔(𝑒^{𝑥}𝑒^{𝑦})

Newman, Morris; Thompson, Robert C.

doi:10.1090/S0025-5718-1987-0866114-9

Numerical values of Goldberg's coefficients in the series for ${\rm log}(e\sp xe\sp y)$

Authors: Morris Newman and Robert C. Thompson
Journal: Math. Comp. 48 (1987), 265-271
MSC: Primary 17B05; Secondary 11Y99, 17-04
DOI: https://doi.org/10.1090/S0025-5718-1987-0866114-9
MathSciNet review: 866114
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Abstract: The coefficients of K. Goldberg in the infinite series for $\log ({e^x}{e^y})$ for noncommuting x and y are computed as far as words of length twenty.

[1] Dragomir Ž. Djoković, An elementary proof of the Baker-Campbell-Hausdorff-Dynkin formula, Math. Z. 143 (1975), no. 3, 209–211. MR 399196, https://doi.org/10.1007/BF01214376
[2] M. Eichler, A new proof of the Baker-Campbell-Hausdorff formula, J. Math. Soc. Japan 20 (1968), 23–25. MR 223417, https://doi.org/10.2969/jmsj/02010023
[3] R. Gilmore, Baker-Campbell-Hausdorff formulas, J. Mathematical Phys. 15 (1974), 2090–2092. MR 354944, https://doi.org/10.1063/1.1666587
[4] Karl Goldberg, The formal power series for 𝑙𝑜𝑔𝑒^{𝑥}𝑒^{𝑦}, Duke Math. J. 23 (1956), 13–21. MR 82571
[5] Marshall Hall Jr., A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950), 575–581. MR 38336, https://doi.org/10.1090/S0002-9939-1950-0038336-7
[6] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
[7] B. V. Lidskiĭ, A polyhedron of the spectrum of the sum of two Hermitian matrices, Funktsional. Anal. i Prilozhen. 16 (1982), no. 2, 76–77 (Russian). MR 659172
[8] Wilhelm Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. 7 (1954), 649–673. MR 67873, https://doi.org/10.1002/cpa.3160070404
[9] Wilhelm Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann. of Math. (2) 52 (1950), 111–126. MR 38964, https://doi.org/10.2307/1969512
[10] R. D. Richtmyer and Samuel Greenspan, Expansion of the Campbell-Baker-Hausdorff formula by computer, Comm. Pure Appl. Math. 18 (1965), 107–108. MR 175292, https://doi.org/10.1002/cpa.3160180111
[11] Olga Taussky and John Todd, Some discrete variable computations, Proc. Sympos. Appl. Math., Vol. 10, American Mathematical Society, Providence, R.I., 1960, pp. 201–209. MR 0115261
[12] R. C. Thompson, Lecture at the 1980 Auburn (Alabama) matrix conference organized by Emilie Haynsworth; three unpublished manuscripts.
[13] Robert C. Thompson, Cyclic relations and the Goldberg coefficients in the Campbell-Baker-Hausdorff formula, Proc. Amer. Math. Soc. 86 (1982), no. 1, 12–14. MR 663855, https://doi.org/10.1090/S0002-9939-1982-0663855-0
[14] Robert C. Thompson, Author vs. referee: a case history for middle level mathematicians, Amer. Math. Monthly 90 (1983), no. 10, 661–668. MR 723938, https://doi.org/10.2307/2323528
[15] Robert C. Thompson, Proof of a conjectured exponential formula, Linear and Multilinear Algebra 19 (1986), no. 2, 187–197. MR 846553, https://doi.org/10.1080/03081088608817715
[16] J. Todd, "Comment on previous self-study answer," Ann. Hist. Comput., v. 7, 1985, p. 69.
[17] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. Prentice-Hall Series in Modern Analysis. MR 0376938