Skip to main content
SpringerLink
Log in

Applications of the Cosecant and Related Numbers

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Power series expansions for cosecant and related functions together with a vast number of applications stemming from their coefficients are derived here. The coefficients for the cosecant expansion can be evaluated by using: (1) numerous recurrence relations, (2) expressions resulting from the application of the partition method for obtaining a power series expansion and (3) the result given in Theorem 3. Unlike the related Bernoulli numbers, these rational coefficients, which are called the cosecant numbers and are denoted by c k , converge rapidly to zero as k→∞. It is then shown how recent advances in obtaining meaningful values from divergent series can be modified to determine exact numerical results from the asymptotic series derived from the Laplace transform of the power series expansion for tcsc (at). Next the power series expansion for secant is derived in terms of related coefficients known as the secant numbers d k . These numbers are related to the Euler numbers and can also be evaluated by numerous recurrence relations, some of which involve the cosecant numbers. The approaches used to obtain the power series expansions for these fundamental trigonometric functions in addition to the methods used to evaluate their coefficients are employed in the derivation of power series expansions for integer powers and arbitrary powers of the trigonometric functions. Recurrence relations are of limited benefit when evaluating the coefficients in the case of arbitrary powers. Consequently, power series expansions for the Legendre-Jacobi elliptic integrals can only be obtained by the partition method for a power series expansion. Since the Bernoulli and Euler numbers give rise to polynomials from exponential generating functions, it is shown that the cosecant and secant numbers gives rise to their own polynomials from trigonometric generating functions. As expected, the new polynomials are related to the Bernoulli and Euler polynomials, but they are found to possess far more interesting properties, primarily due to the convergence of the coefficients. One interesting application of the new polynomials is the re-interpretation of the Euler-Maclaurin summation formula, which yields a new regularisation formula.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Segur, H., Tanveer, S., Levine, H. (eds.): Asymptotics Beyond All Orders. Plenum Press, New York (1991)

    MATH  Google Scholar 

  2. 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)

    MATH  Google Scholar 

  3. Boyd, J.P.: The Devil’s invention: asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Math. 56, 1–98 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kowalenko, V.: The Stokes Phenomenon, Borel Summation and Mellin-Barnes Regularisation. Bentham e-books, The Netherlands (2009)

    Google Scholar 

  5. Kowalenko, V.: Properties and applications of the reciprocal logarithm numbers. Acta Appl. Math. 109, 413–427 (2009). doi:10.1007/s10440-008-9325-0

    Article  MathSciNet  Google Scholar 

  6. Kowalenko, V.: Generalizing the reciprocal logarithm numbers by adapting the partition method for a power series expansion. Acta Appl. Math. 106, 369–420 (2008). doi:10.1007/s10440-008-9304-5

    Article  MathSciNet  Google Scholar 

  7. Wolfram, S.: Mathematica—A System for Doing Mathematics by Computer. Addison-Wesley, Reading (1992)

    Google Scholar 

  8. 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 

  9. Kowalenko, V.: Exactification of the asymptotics for Bessel and Hankel functions. Appl. Math. Comput. 133, 487–518 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn., p. 252. Cambridge University Press, Cambridge (1973)

    Google Scholar 

  11. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1965)

    Google Scholar 

  12. Knuth, D.E.: The Art of Computer Programming. Fascicle 3: Generating All Combinations and Partitions, vol. IV. Addison-Wesley, New York (2005)

    Google Scholar 

  13. McKay, J.K.S.: Algorithm 371 partitions in natural order [A1]. Commun. ACM 13, 52 (1970)

    Article  Google Scholar 

  14. Gradshteyn, I.S., Ryzhik, I.M., Jeffrey, A. (eds.): Table of Integrals, Series and Products, 5th edn. Academic Press, London (1994)

    MATH  Google Scholar 

  15. Bierens de Haan, D.: Nouvelles Tables d’Intégrales Définies. Leide, P. Engels (1867)

    Google Scholar 

  16. Knopp, K.: Theory and Application of Infinite Series, 2nd edn. Blackie & Son, London (1959)

    Google Scholar 

  17. Weisstein, E.W.: Cosecant, MathWorld-A wolfram web resource. http://mathworld.wolfram.com/Cosecant.html

  18. Sloane, N.J.: Integer Sequences A036280 and 36281 in: The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org (2010)

  19. Knuth, D.E.: The Art of Computer Programming, vol. 2, Semi-Numerical Algortihms, 3rd edn. Addison-Wesley, Reading (1997)

    Google Scholar 

  20. Prudnikov, A.P., Brychkov, Yu.A., Marichev, O.I.: Integrals and Series. Elementary Functions, vol. I. Gordon & Breach, New York (1986)

    Google Scholar 

  21. Dingle, R.B.: Asymptotic Expansions: Their Derivation and Interpretation. Academic Press, London (1973)

    MATH  Google Scholar 

  22. Luke, Y.L.: The Special Functions and Their Approximations, vol. I. Academic Press, New York (1969)

    MATH  Google Scholar 

  23. Oberhettinger, F.: Tables of Mellin Transforms. Springer, Berlin (1974)

    MATH  Google Scholar 

  24. Berry, M.V.: Uniform asymptotic smoothing of Stokes’s discontinuities. Proc. R. Soc. Lond. A 422, 7–21 (1989)

    Article  MATH  Google Scholar 

  25. Apelblat, A.: Tables of Integrals and Series. Verlag Harri Deutsch, Thun (1996)

    Google Scholar 

  26. Pickover, C.A.: The Mathematics of Oz: Mental Gymnastics from Beyond the Edge, pp. 57–59, 265–268. Cambridge University Press, New York (2002)

    MATH  Google Scholar 

  27. Weisstein, E.W.: Flint Hills series, MathWorld-A wolfram web resource. http://mathworld.wolfram.com/FlintHillsSeries.html

  28. Spanier, J., Oldham, K.B.: An Atlas of Functions. Hemisphere Publishing, New York (1987)

    MATH  Google Scholar 

  29. Wheelon, A.D.: Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco (1968)

    MATH  Google Scholar 

  30. Klamkin, M.S. (ed.): Problems in Applied Mathematics—Selections from the SIAM Review. SIAM, Philadelphia (1990)

    Google Scholar 

  31. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge Mathematical Library, Cambridge (1995), Ch. XIX

    MATH  Google Scholar 

  32. Mangulis, V.: Handbook of Series for Scientists and Engineers. Academic Press, New York (1965)

    MATH  Google Scholar 

  33. Hansen, E.: A Table of Series and Products. Prentice-Hall, Englewood Cliffs (1975)

    MATH  Google Scholar 

  34. Press, W.H., et al.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, New York (1992)

    Google Scholar 

  35. Copson, E.T.: An Introduction to the Theory of Functions of a Complex Variable. Clarendon Press, Oxford (1973)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ARC Centre for Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Melbourne, Victoria, 3010, Australia

    Victor Kowalenko

Authors
  1. Victor Kowalenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Kowalenko.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kowalenko, V. Applications of the Cosecant and Related Numbers. Acta Appl Math 114, 15–134 (2011). https://doi.org/10.1007/s10440-011-9604-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-011-9604-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation