The Ramanujan Journal

, Volume 15, Issue 2, pp 177–185 | Cite as

Expression for restricted partition function through Bernoulli polynomials

Article

Abstract

Explicit expressions for restricted partition function W(s,d m ) and its quasiperiodic components W j (s,d m ) (called Sylvester waves) for a set of positive integers d m ={d 1,d 2,…,d m } are derived. The formulas are represented in a form of a finite sum over Bernoulli polynomials of higher order with periodic coefficients.

Keywords

Restricted partitions Bernoulli polynomials of higher order 

Mathematics Subject Classification (2000)

11P81 11B68 11B37 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrews, G.E.: The Theory of Partitions. Encyclopedia of Mathematics and Its Applications, vol. 2. Addison-Wesley, Reading (1976) MATHGoogle Scholar
  2. 2.
    Bateman, H., Erdelýi, A.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953) Google Scholar
  3. 3.
    Beck, M., Gessel, I.M., Komatsu, T.: The polynomial part of a restriction partition function related to the Frobenius problem. Electr. J. Comb. 8(7), 1–5 (2001) MathSciNetGoogle Scholar
  4. 4.
    Carlitz, L.: Eulerian numbers and polynomials of higher order. Duke Math. J. 27, 401–423 (1960) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cayley, A.: Researches on the partitions of numbers. Phil. Trans. Roy Soc. 145, 127–140 (1855) Google Scholar
  6. 6.
    Comtet, L.: Advanced Combinatorics, Chapter 2. Reidel, Dordrecht (1974) Google Scholar
  7. 7.
    Dickson, L.: History of the Theory of Numbers, vol. 2, Chapter 3. Chelsea, New York (1976) Google Scholar
  8. 8.
    Frobenius, F.G.: Über die Bernoullischen Zahlen und die Eulerischen Polynome, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 809–847 (1910) Google Scholar
  9. 9.
    Nörlund, N.E.: Mémoire sur les polynomes de Bernoulli. Acta Math. 43, 121–196 (1922) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Riordan, J.: An Introduction to Combinatorial Analysis. Wiley Publications in Mathematical Statistics. Wiley/Chapman and Hall, London (1958) MATHGoogle Scholar
  11. 11.
    Roman, S., Rota, G.-C.: The umbral calculus. Adv. Math. 27, 95–188 (1978) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Rubinstein, B.Y., Fel, L.F.: Restricted partition function as Bernoulli and Eulerian polynomials of higher order. Ramanujan J. 11, 331–347 (2006) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sylvester, J.J.: On the partition of numbers. Q. J. Math. 1, 141–152 (1857) Google Scholar
  14. 14.
    Sylvester, J.J.: On subinvariants, i.e. semi-invariants to binary quantics of an unlimited order with an excursus on rational fractions and partitions. Am. J. Math. 5, 79–136 (1882) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Stowers Institute for Medical ResearchKansas CityUSA

Personalised recommendations