A Symbolic Treatment of Abel Polynomials

Contributed Chapter
  • Pasquale Petrullo


We study the umbral polynomials A(k) n(x, α) = x(xk·α) n−1, by means of which a wide range of formal power series identities, including Lagrange inversion formula, can be usefully manipulated. We apply this syntax within cumulant theory, and show how moments and its formal cumulants (classical, free and Boolean) are represented by polynomials A(k) n(α, γ) for suitable choices of umbrae α and γ.


Formal Power Series Noncrossing Partition Parking Function Free Convolution Free Probability Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Di Nardo, E. Niederhausen, H. Senato, D., The Classical Umbral Calculus: Sheffer Sequences. arXiv:0810. 3554v1.Google Scholar
  2. 2.
    Di Nardo, E., Petrullo, and P., Senato, D., Cumulants, Convolutions and Volume Polynomial, preprint.Google Scholar
  3. 3.
    Di Nardo, E., Senato, D. (2006), An Umbral Setting for Cumulants and Factorial Moments, in “European J. Combin.”, 27, pp. 394–41.Google Scholar
  4. 4.
    Di Nardo, E., Senato, D. (2009), The Problems Eleven and Twelve of Rota’s Fubini Lectures: from Cumulants to Free Probability Theory, published in this volume.Google Scholar
  5. 5.
    Doubilet, P., Rota, G.-C., Stanley, R. P. (1972), On the Foundations of Combinatorial Theory (VI): The Idea of Generating Function, in Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. II: Probability Theory, Berkeley, University of California, pp. 267–318.Google Scholar
  6. 6.
    Lehner, F. (2002), Free Cumulants and Enumeration of Connected Partitions, in “European J. Combin.”, 23, pp. 1025–31.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Nica, A., Speicher, R. (1997), A “Fourier Transform” for Multiplicative Functions on Non-crossing Partitions, in “J. Algebraic Combin.”, 6, pp. 141–60.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Petrullo, P., Senato, D., An Instance of Umbral Methods in Representation Theory: the Parking Function Module, in “Pure Math. Appl.”, to be published.Google Scholar
  9. 9.
    Roman, S. (1984), The Umbral Calculus, S. Eilenberg, H. Bass (eds.), London, Academic Press.MATHGoogle Scholar
  10. 10.
    Rota, G.-C., Shen, J. (2000), On the Combinatorics of Cumulants, in “J. Combin. Theory Ser. A”, 91, pp. 283–304.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rota, G.-C., Shen, J., Taylor, B. D. (1997), All Polynomials of Binomial Type are Represented by Abel Plynomials, in “Ann. Scuola Norm. Sup. Pisa Cl. Sci.”, 25/1, pp. 731–8.MathSciNetGoogle Scholar
  12. 12.
    Rota, G.-C., Taylor, B. D. (1994), The Classical Umbral Calculus, in “SIAM J. Math. Anal.”, 25, pp. 694–711.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Speicher, R. (1994), Multiplicative Functions on the Lattice of Non-crossing Partitions and Free Convolution, in “Math. Ann.”, 298, pp. 611–28.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Stanley, R. P. (1999), Enumerative Combinatorics, vol. 2, Cambridge, Cambridge University Press.CrossRefGoogle Scholar
  15. 15.
    Speicher, R., Woroudi, R. (1997), Boolean Convolution, in Free Probability Theory (Waterloo, ON, 1995), Providence, RI, American Mathematical Society, pp. 267–79.Google Scholar
  16. 16.
    Voiculescu, D. (2000), Lectures on Free Probability Theory, in Lectures on Probability Theory and Statistics, Berlin, Springer, pp. 279–349.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Pasquale Petrullo
    • 1
  1. 1.Università degli Studi della BasilicataBasilicataItaly

Personalised recommendations