Skip to main content
SpringerLink
Log in

Decomposition and Group Theoretic Characterization of Pairs of Inverse Relations of the Riordan Type

  • Published:
Acta Applicandae Mathematica Aims and scope Submit manuscript

Abstract

A new solution to Riordan’s problem of combinatorial identities classification is presented. An algebgraic characterization of pairs of inverse relations of the Riordan type is given. The use of the integral representation approach for generating new types of combinatorial identities is demonstrated.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barcucci, E., Pinzani, R. and Sprugnoli, R.: The random generation of under diagonal walks, Discrete Math. 139 (1995), 3–19.

    Google Scholar 

  2. Bender, E. A.: Central and local limit theorems applied to asymptotic enumeration, J. Combin. Theory, Ser. A 15 (1973), 91–111.

    Google Scholar 

  3. Cartan, H.: Théorie élémentaire des fonctions analytiques d’une on plusieurs variables complexes, Hermann, Paris, 1961.

    Google Scholar 

  4. Dingle, R. B.: Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, New York, 1974.

    Google Scholar 

  5. Egorychev, G. P.: Integral Representation and the Computation of Combinatorial Sums, Nauka, Novosibirsk, 1977 (in Russian); Transl. of Math. Monographs, Vol. 59, Amer. Math. Soc., 1984, 2nd edn in 1989 (in English).

    Google Scholar 

  6. Egorychev, G. P.: Inversion of one-dimensional combinatorial relations, In: Some Questions on the Theory of Groups and Rings, Inst. Fiz. Sibirsk. Otdel. Akad. Nauk SSSR, Krasnoyarsk, 1973, pp. 110–122 (in Russian).

    Google Scholar 

  7. Egorychev, G. P.: The inversion of combinatorial relations, Combinatorial Analysis (Krasn. State University) 3 (1974), 10–14 (in Russian).

    Google Scholar 

  8. Egorychev, G. P.: Algorithms of integral representation of combinatorial sums and their applications. Formal power series and algebraic combinatorics, In: Proceedings of 12th International Conference, FPSAC’00, Moscow, Russia, June 2000, 2000, pp. 15–29.

  9. Gessel, I. M.: A noncommutative generalization and q-analog of the Lagrange inversion formula, Trans. Amer. Math. Soc. 257 (1980), 455–482.

    Google Scholar 

  10. Goulden, I. P. and Jackson, D. M.: Combinatorial Enumeration, Willey, New York, 1983.

    Google Scholar 

  11. Hardy, G. H.: Divergent Series, Clarendon Press, Oxford, 1949.

    Google Scholar 

  12. Henrici, P.: Applied and Computational Complex Analysis, Wiley, New York, 1991.

    Google Scholar 

  13. Jabotinski, E.: Sur la representation de la composition de fonctions par un product de matrices, Comptes Rendus 224 (1947), 323–324.

    Google Scholar 

  14. Jabotinski, E.: Analytic iterations, Trans. Amer. Math. Soc. 108 (1963), 457–477.

    Google Scholar 

  15. Krattenthaler, Ch.: A new q-Lagrange formula and some applications, Proc. Amer. Math. Soc. 90 (1984), 338–344.

    Google Scholar 

  16. Krattenthaler, Ch.: Operator methods and Laplace inversion, a unified approach to Lagrange formulas, Trans. Amer. Math. Soc. 305 (1988), 431–465.

    Google Scholar 

  17. Krattenthaler, Ch.: A new matrix inverse, Proc. Amer. Math. Soc. 124 (1996), 47–59.

    Google Scholar 

  18. Krattenthaler, Ch. and Schlosser, M.: A new matrix inverse with applications to multiple q-series, Discrete Math. 204 (1999), 249–279.

    Google Scholar 

  19. Merlini, D., Rogers, D. S., Sprugnoli, R. and Verbi, C.: On some alternative characterizations of Riordan groups, Canad. J. Math. 49(2) (1997), 301–320.

    Google Scholar 

  20. Merlini, D. and Verri, C.: Generating trees and proper Riordan Arrays, Discrete Math. 218 (2000), 167–183.

    Google Scholar 

  21. Peart, P. and Woan Wen-Jin: A divisibility property for a subgroup of Riordan matrices, Discrete Appl. Math. 98 (2000), 256–263.

    Google Scholar 

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

    Google Scholar 

  23. Riordan, J.: Combinatorial Identities, Wiley, New York, 1968.

    Google Scholar 

  24. Riordan, J.: Inverse relations and combinatorial identities, Amer. Math. Monthly 71 (1964), 485–498.

    Google Scholar 

  25. Roman, S.: The algebra of formal series, Adv. in Math. 31 (1979), 309–329.

    Google Scholar 

  26. Shapiro, L. W., Getu, S., Woan Wen-Jin and Woodson, L. C.: The Riordan group, Discrete Appl. Math. 34 (1991), 229–239.

    Google Scholar 

  27. Sloane, N. J. A. and Plouffe, S.: The Encyclopedia of Integer Sequences, Academic Press, San Diego, 1995.

    Google Scholar 

  28. Sprugnoli, R.: Riordan arrays and combinatorial sums, Discrete Math. 132 (1994), 267–290.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Krasnoyarsk State Technical University, Kirenskogo 26, Krasnoyarsk, 660074, Russia

    Georgy P. Egorychev

  2. University of Waterloo, 200 University Ave. W., Waterloo, Ontario, Canada, N2L 3G1

    Eugene V. Zima

Authors
  1. Georgy P. Egorychev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eugene V. Zima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georgy P. Egorychev.

Additional information

Supported in part by the National Sciences and Engineering Research Council of Canada on Grant NSERC-108343.

Mathematics Subject Classifications (2000)

combinatorics, algebra.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Egorychev, G.P., Zima, E.V. Decomposition and Group Theoretic Characterization of Pairs of Inverse Relations of the Riordan Type. Acta Appl Math 85, 93–109 (2005). https://doi.org/10.1007/s10440-004-5589-1

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-004-5589-1

Keywords

Search

Navigation