zbMATH — the first resource for mathematics

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 \(1/\arctan x\).

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
Mathematica; OEIS
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.