<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume10 Issue2" %>
The Asymptotic Behavior of Certain Birth Processes
Siddhartha Sahi
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA
Annals of Combinatorics 10 (2) p. 255-269 June, 2006
AMS Subject Classification: 60J80, 60J85, 05A19, 33D52
We describe a connection between discrete birth process and a certain family of multivariate interpolation polynomials. This enables us to compute all asymptotic moments of the birth process, generalizing previously known results for the mean and variance.
Keywords: birth processes, cumulants, Capelli identity, interpolation polynomials, divisors, q¨Cseries


1. G. Andrews, D. Crippa, and K. Simon, q-Series arising from the study of random graphs, SIAM J. Discrete Math. 10 (1997) 41–56.

2. T. Bickel, N. Galli, and K. Simon, Birth processes and symmetric polynomials, Ann. Combin. 5 (2001) 123–139.

3. K. Dilcher, Some q-series identities related to divisor functions, Discrete Math. 145 (1995) 83–93.

4. W. Feller, An Introduction to Probability Theory and Its Applications, 2nd Ed., John Wiley and Sons, Inc., New York-London-Sydney, 1971.

5. A.M. Fu and A. Lascoux, q-Identities from Lagrange and Newton interpolation, Adv. Appl. Math. 31 (2003) 527–531.

6. F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeros, Internat. Math. Res. Notices 10 (1996) 473–486.

7. I. Macdonald, Symmetric Functions and Hall Polynomials, 2nd ed., Oxford University Press, 1995.

8. H. Prodinger, Some applications of the q-Rice formula, Random Structures Algorithms 19 (2001) 552–557.

9. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966.

10. S. Sahi, The spectrum of certain invariant differential operators associated to Hermitian symmetric spaces, Progr. Math. 123 (1994) 569–576.

11. S. Sahi, Interpolation, integrality and a generalization of Macdonald's polynomials, Internat. Math. Res. Notices 10 (1996) 457–471.

12. K. Uchimura, Divisor generating functions and insertion into a heap, Discrete Appl. Math. 18 (1987) 73–81.

13. E. Whittaker and G. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, 1927.