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Asymptotics of generalized derangements

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Abstract

We derive the asymptotics of certain combinatorial numbers defined on multi-sets when the number of sets tends to infinity but the sizes of the sets remain fixed. This includes the asymptotics of generalized derangements, numbers related to k-partite graphs, and exponentially weighted derangements. The asymptotics use integral and sum representations of the numbers involved. We also explore the combinatorial implications of the asymptotic results. In fact we first derive general asymptotic formulas for integrals and sums of certain types and then we specialize them to study the asymptotics of the combinatorial numbers.

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References

  1. Andrews, G., Askey, R.A., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  2. Artin, E.: The Gamma Function. Holt, Rinehart and Winston, New York (1964)

    MATH  Google Scholar 

  3. Askey, R.A.: Orthogonal Polynomials and Special Functions. Society for Industrial and Applied Mathematics, Philadelphia, PA (1975)

    Book  Google Scholar 

  4. Askey, R.A., Ismail, M.E.H.: Permutation problems and special functions. Can. J. Math. 28, 853–874 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  5. Askey, R.A., Ismail, M.E.H., Koornwinder, T.: Weighted permutation problems and Laguerre polynomials. J. Comb. Theory Ser. A 25, 277–287 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Azor, R., Gillis, J., Victor, J.D.: Combinatorial applications of Hermite polynomials. SIAM J. Math. Anal. 13, 879–890 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)

  8. Even, S., Gillis, J.: Derangements and Laguerre polynomials. Math. Proc. Camb. Philos. Soc. 79, 135–143 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  9. Foata, D., Zeilberger, D.: Laguerre polynomials, weighted derangements and positivity. SIAM J. Discrete Math. 1, 425–433 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  11. Gillis, J., Ismail, M.E.H., Offer, T.: An asymptotic problem in derangement theory. SIAM J. Math. Anal. 21, 262–269 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge (2005)

    Book  MATH  Google Scholar 

  13. Ismail, M.E.H., Stanton, D., Viennot, G.: The combinatorics of the q-Hermite polynomials and the Askey–Wilson integral. Eur. J. Comb. 8, 379–392 (1987)

    MathSciNet  MATH  Google Scholar 

  14. Ismail, M.E.H., Kasraoui, A., Zeng, J.: Separation variables and linearization coefficients of orthogonal polynomials. Adv. Math (2011, submitted)

  15. Kim, D.S., Zeng, J.: A combinatorial formula for the linearization coefficients of general Sheffer polynomials. Eur. J. Comb. 22, 313–332 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Koekoek, R., Swarttouw, R.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogues. Reports of the Faculty of Technical Mathematics and Informatics, No. 98-17. Delft University of Technology, Delft (1998)

  17. Ksavrelof, G., Zeng, J.: Two involutions for signed excedance numbers. Sémin. Lothar. Comb. 49, Article B49e (2003)

  18. MacMahon, P.: Combinatory Analysis, vols. 1 and 2. Cambridge University Press, Cambridge (1915–1916). Reprinted by Chelsea, New York (1984)

  19. Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974)

    Google Scholar 

  20. Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge University Press (1999)

  21. Rainville, E.D.: Special Functions. Macmillan, New York (1960)

    MATH  Google Scholar 

  22. Riordan, J.: Introduction to Combinatorial Analysis. Wiley, New York (1958)

    MATH  Google Scholar 

  23. Szegő, G.: Orthogonal Polynomials, 4th edn. Amer. Math. Soc., Providence (1975)

  24. Viennot, X.G.: Une Théorie Combinatoire des Polynômes Orthogonaux Généraux. Lecture Notes. Université du Québec à Montréal, Montreal (1983)

  25. Wilf, H.S.: Generatingfunctionology, 3rd edn. A K Peters, Wellesley, MA (2006)

  26. Whittaker, E.T., Watson, G.N.: A Course in Modern Analysis. Cambridge Mathematical Library (1996)

  27. Zeng, J.: Linéarisation de produits de polynômes de Meixner, Krawtchouk, et Charlier. SIAM J. Math. Anal. 21, 1349–1368 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zeng, J.: Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. Lond. Math. Soc. 65, 1–22 (1992)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. City University of Hong Kong, Kowloon, Hong Kong

    Mourad E. H. Ismail

  2. King Saud University, Riyadh, Kingdom of Saudi Arabia

    Mourad E. H. Ismail

  3. Department of Computer and Mathematical Sciences, University of Houston-Downtown, Houston, TX, 77002, USA

    Plamen Simeonov

Authors
  1. Mourad E. H. Ismail
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  2. Plamen Simeonov
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Corresponding author

Correspondence to Plamen Simeonov.

Additional information

Communicated by Charles Micchelli.

This research is supported by the Research Grants Council of Hong Kong under contract # 101410. The research of Mourad E. H. Ismail is also supported by the NPST Program of King Saud University, Project Number 10-MAT1293-02, and a grant from King Saud University, Kingdom of Saudi Arabia.

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Ismail, M.E.H., Simeonov, P. Asymptotics of generalized derangements. Adv Comput Math 39, 101–127 (2013). https://doi.org/10.1007/s10444-011-9271-7

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  • DOI: https://doi.org/10.1007/s10444-011-9271-7

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