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Properties and applications of the reciprocal logarithm numbers. (English) Zbl 1208.11032
The author investigates the sequence of rational numbers $$(A_k)_{k\geq 0}$$, which he calls the “reciprocal logarithm numbers” defined by the series expansion $$1/\ln(1+z) = \sum_{k=0}^\infty A_k z^{k-1}$$. He derives several properties of these numbers such as recursions, integral representations and identities. More specifically, he points out the relationship of the $$A_k$$’s to Stirling numbers of the first kind and to Euler’s constant. A similar approach is then presented for the series expansion of $$1/\arctan x$$.

##### MSC:
 11B73 Bell and Stirling numbers 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) 05A17 Combinatorial aspects of partitions of integers 05C90 Applications of graph theory
##### Software:
Mathematica; OEIS
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##### References:
 [1] Kowalenko, V., Frankel, N.E.: Asymptotics for the Kummer function of Bose plasmas. J. Math. Phys. 35, 6179–6198 (1994) · Zbl 0815.33013 [2] Wolfram, S.: Mathematica–A System for Doing Mathematics by Computer. Addison-Wesley, Reading (1992) · Zbl 0925.65002 [3] Kowalenko, V.: Towards a theory of divergent series and its importance to asymptotics. In: Recent Research Developments in Physics, vol. 2, pp. 17–68. Transworld Research Network, Trivandrum (2001) [4] Kowalenko, V.: Exactification of the asymptotics for Bessel and Hankel functions. Appl. Math. Comput. 133, 487–518 (2002) · Zbl 1070.33004 [5] Kowalenko, V., Frankel, N.E., Glasser, M.L., Taucher, T.: Generalised Euler-Jacobi Inversion Formula and Asymptotics beyond All Orders. London Mathematical Society Lecture Note, vol. 214. Cambridge University Press, Cambridge (1995) · Zbl 0856.33003 [6] Kowalenko, V.: The non-relativistic charged Bose gas in a magnetic field II. Quantum properties. Ann. Phys. (N.Y.) 274, 165–250 (1999) · Zbl 1034.82514 [7] Weisstein, E.W.: Logarithmic number. MathWorld–A Wolfram Web Resource, http://mathworld.wolfram.com/LogarithmicNumber.html [8] Sloane, N.J.A.: The on-line encyclopedia of integer sequences, http://www.research.att.com/njas/seequences · Zbl 1274.11001 [9] Kowalenko, V.: The Stokes phenomenon, Borel summation and Mellin-Barnes regularisation. To be published by Bentham e-books · Zbl 1342.41001 [10] Spanier, J., Oldham, K.B.: An Atlas of Functions. Hemisphere Publishing, New York (1987) · Zbl 0618.65007 [11] Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn., p. 252. Cambridge University Press, Cambridge (1973) · JFM 45.0433.02 [12] Knuth, D.E.: The Art of Computer Programming, vol. 2. Addison-Wesley, Reading (1981) · Zbl 0477.65002 [13] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965) · Zbl 0171.38503 [14] Copson, E.T.: An Introduction to the Theory of Functions of a Complex Variable, p. 24. Clarendon Press, Oxford (1976) · Zbl 0188.37901 [15] Weisstein, E.W., et al.: Harmonic number. Mathworld–A Wolfram Web Resource, http://mathworld.wolfram.com//HarmonicNumber.html [16] Comtet, L.: Advanced Combinatorics. Reidel, Dordrecht (1974) · Zbl 0283.05001 [17] Adamchik, V.: On Stirling numbers and Euler sums. J. Comput. Appl. Math. 79, 119–130 (1997) · Zbl 0877.39001 [18] Prudnikov, A.P., Marichev, O.I., Brychkov, Yu.A.: Elementary Functions. Integrals and Series, vol. I. Gordon and Breach, New York (1986) · Zbl 0733.00004 [19] Gradshteyn, I.S., Ryzhik, I.M., Jeffrey, A. (eds.): Table of Integrals, Series and Products, 5th edn. Academic Press, London (1994) · Zbl 0918.65002 [20] Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen, 2nd edn. Chelsea, New York (1953) · Zbl 0051.28007
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