Acta Applicandae Mathematicae

, Volume 109, Issue 2, pp 413–437 | Cite as

Properties and Applications of the Reciprocal Logarithm Numbers

  • Victor KowalenkoEmail author


Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the reciprocal of the logarithmic function ln (1+z), whose coefficients represent an infinite set of fractions. These numbers, which are called reciprocal logarithm numbers and are denoted by A k , converge to zero as k→∞. Several properties of the numbers are then obtained including recursion relations and their relationship with the Stirling numbers of the first kind. Also appearing here are several applications including a new representation for Euler’s constant known as Hurst’s formula and another for the logarithmic integral. From the properties of the A k it is found that a term of ζ(2) cannot be eliminated by the remaining terms in Hurst’s formula, thereby indicating that Euler’s constant is irrational. Finally, another power series expansion for the reciprocal of arctangent is developed by adapting the preceding material.


Convergence Divergent series Equivalence Euler’s constant Harmonic numbers Hurst’s formula Partitions Power series expansions Reciprocal logarithm numbers Recursion relation Regularization Soldner’s constant Stirling numbers of the first kind Tree diagram 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kowalenko, V., Frankel, N.E.: Asymptotics for the Kummer function of Bose plasmas. J. Math. Phys. 35, 6179–6198 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Wolfram, S.: Mathematica—A System for Doing Mathematics by Computer. Addison-Wesley, Reading (1992) Google Scholar
  3. 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) Google Scholar
  4. 4.
    Kowalenko, V.: Exactification of the asymptotics for Bessel and Hankel functions. Appl. Math. Comput. 133, 487–518 (2002) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 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) zbMATHGoogle Scholar
  6. 6.
    Kowalenko, V.: The non-relativistic charged Bose gas in a magnetic field II. Quantum properties. Ann. Phys. (N.Y.) 274, 165–250 (1999) zbMATHCrossRefGoogle Scholar
  7. 7.
    Weisstein, E.W.: Logarithmic number. MathWorld—A Wolfram Web Resource,
  8. 8.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences,
  9. 9.
    Kowalenko, V.: The Stokes phenomenon, Borel summation and Mellin-Barnes regularisation. To be published by Bentham e-books Google Scholar
  10. 10.
    Spanier, J., Oldham, K.B.: An Atlas of Functions. Hemisphere Publishing, New York (1987) zbMATHGoogle Scholar
  11. 11.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn., p. 252. Cambridge University Press, Cambridge (1973) Google Scholar
  12. 12.
    Knuth, D.E.: The Art of Computer Programming, vol. 2. Addison-Wesley, Reading (1981) zbMATHGoogle Scholar
  13. 13.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965) Google Scholar
  14. 14.
    Copson, E.T.: An Introduction to the Theory of Functions of a Complex Variable, p. 24. Clarendon Press, Oxford (1976) Google Scholar
  15. 15.
    Weisstein, E.W., et al.: Harmonic number. Mathworld—A Wolfram Web Resource,
  16. 16.
    Comtet, L.: Advanced Combinatorics. Reidel, Dordrecht (1974) zbMATHGoogle Scholar
  17. 17.
    Adamchik, V.: On Stirling numbers and Euler sums. J. Comput. Appl. Math. 79, 119–130 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Prudnikov, A.P., Marichev, O.I., Brychkov, Yu.A.: Elementary Functions. Integrals and Series, vol. I. Gordon and Breach, New York (1986) zbMATHGoogle Scholar
  19. 19.
    Gradshteyn, I.S., Ryzhik, I.M., Jeffrey, A. (eds.): Table of Integrals, Series and Products, 5th edn. Academic Press, London (1994) zbMATHGoogle Scholar
  20. 20.
    Landau, E.: Handbuch der Lehre von der Verteilung der Primzahlen, 2nd edn. Chelsea, New York (1953) zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of PhysicsUniversity of MelbourneParkvilleAustralia

Personalised recommendations