Skip to main content

Enumeration and Special Functions

  • Chapter
  • First Online:
Orthogonal Polynomials and Special Functions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1817))

  • 1416 Accesses

Abstract

These notes for the August 12–16, 2002 Euro Summer School in OPSF at Leuven have three sections with basic introductions to

  1. 1

    Enumeration and q -series

  2. 2

    Enumeration and orthogonal polynomials

  3. 3

    Symmetric functions. No prior exposure to these areas is assumed. Three excellent textbooks for these three topics are [1], [7], and [17]. Several exercises and open problems are given throughout these notes.

Research partially supported by NSF grant DMS 0203282

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2 (G.-C. Rota ed.), Addison-Wesley, Reading, Mass., 1976 (reissued by Cambridge Univ. Press, London and New York, 1985).

    Google Scholar 

  2. ___, On a conjecture of Peter Borwein, J. Symbolic Computation 20 (1995), 487–501.

    Article  MATH  Google Scholar 

  3. ___, On the proofs of the Rogers-Ramanujan identities, in ‘q-Series and Partitions’, IMA Vol. Math. 18, Springer, New York, 1989, pp. 1–14.

    Google Scholar 

  4. G. Andrews, R. Baxter, D. Bressoud, W. Burge, P. Forrester, G. Viennot, Partitions with prescribed hook differences, Eur. J. Comb. 8 (1987), 341–350.

    MATH  MathSciNet  Google Scholar 

  5. D. Bressoud, The Borwein conjecture and partitions with prescribed hook differences, Elec. J. Comb. 3 (1996), 14 pp.

    MathSciNet  Google Scholar 

  6. ___, Proofs and confirmations. The story of the alternating sign matrix conjecture, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  7. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

    MATH  Google Scholar 

  8. P. Flajolet, Combinatorial aspects of continued fractions, Disc.Math. 32 (1980), 125–161.

    MATH  MathSciNet  Google Scholar 

  9. D. Foata, A combinatorial proof of the Mehler formula, J. Comb. A 24 (1978), 367–376.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Foata and P. Leroux, Polynômes de Jacobi, interpretation combinatoire et fonction génératrice, Proc. Amer. Math. Soc. 87 (1983), 47–53.

    Google Scholar 

  11. D. Foata and V. Strehl, Combinatorics of Laguerre polynomials, in ‘Enumeration and Design’, Academic Press, Toronto, 1984, pp. 123–140.

    Google Scholar 

  12. A. Garsia and S. Milne, Method for constructing bijections for classical partition identities, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), 2026–2028.

    Google Scholar 

  13. F. Garvan, D. Kim, and D. Stanton, Cranks and t-cores, Inv.Math. 101 (1990), 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.

    MATH  Google Scholar 

  15. I. Gessel and G. Viennot, Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1985), 300–321.

    Article  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  17. I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, Oxford, 1995.

    MATH  Google Scholar 

  18. ___, Affine root systems and Dedekind’s η-function, Inv. Math. 15 (1972), 91–143.

    Google Scholar 

  19. K. O’Hara, Unimodality of Gaussian coefficients: a constructive proof, J. Comb. A 53 (1990), 29–52.

    Article  MATH  MathSciNet  Google Scholar 

  20. I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, reprinted in ‘I. Schur, Gesammelte Abhandlungen’, volume 2, Springer, Berlin, 1973, pp. 117–136.

    Google Scholar 

  21. R. Simion and D. Stanton, Octabasic Laguerre polynomials and permutation statistics, J. Comput. Appl. Math. 68 (1996), 297–329.

    Article  MATH  MathSciNet  Google Scholar 

  22. R. Stanley, Enumerative Combinatorics, Wadsworth, Monterey, 1986.

    Google Scholar 

  23. ___, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Alg. Disc. Methods 1 (1980), 168–183.

    Google Scholar 

  24. D. Stanton, Gaussian integrals and the Rogers-Ramanujan identities, in ‘Symbolic computation, number theory, special functions, physics, and combinatorics’ (F. Garvan and M. Ismail, eds.), Kluwer, Dordrecht, 2001, pp. 255–266.

    Google Scholar 

  25. G. Viennot, Une Théorie Combinatoire des Polynômes Orthogonaux Généraux, Lecture Notes, University of Quebec at Montreal, 1983.

    Google Scholar 

  26. D. Zeilberger, A one-line high school algebra proof of the unimodality of the Gaussian polynomials [n k]for k 6lt; 20, in ‘q-Series and Partitions’, IMA Vol. Math. 18, Springer, New York, 1989, pp. 67–72.

    Google Scholar 

  27. ___, A q-Foata proof of the q-Saalschütz identity, Eur. J. Comb. 8 (1987), 461–463.

    MATH  MathSciNet  Google Scholar 

  28. ___, Proof of the alternating sign matrix conjecture, Elec. J. Comb. 3 (1996), 1–84.

    Google Scholar 

  29. D. Zeilberger and D. Bressoud, A proof of Andrews’ q-Dyson conjecture, Disc. Math. 54 (1985), 201–224.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Stanton, D. (2003). Enumeration and Special Functions. In: Koelink, E., Van Assche, W. (eds) Orthogonal Polynomials and Special Functions. Lecture Notes in Mathematics, vol 1817. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44945-0_4

Download citation

  • DOI: https://doi.org/10.1007/3-540-44945-0_4

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40375-3

  • Online ISBN: 978-3-540-44945-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics